Construction and Analysis of Multi-Rate Partitioned Runge-Kutta Methods

Transcription

1 Author(s) Mugg, Patrck R. Ttle Costructo ad Aalyss of Mult-Rate Parttoed Ruge-Kutta Methods Publsher Moterey, Calfora. Naval Postgraduate School Issue Date 0-06 URL Ths documet was dowloaded o July 8, 03 at 7:5:35

2 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS CONSTRUCTION AND ANALYSIS OF MULTI-RATE PARTITIONED RUNGE-KUTTA METHODS by Patrck R. Mugg Jue 0 Thess Advsor: Secod Reader: Fracs Graldo Hog Zhou Approved for publc release; dstrbuto s ulmted

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4 REPORT DOCUMENTATION PAGE Form Approved OMB No Publc reportg burde for ths collecto of formato s estmated to average hour per respose, cludg the tme for revewg structo, searchg exstg data sources, gatherg ad matag the data eeded, ad completg ad revewg the collecto of formato. Sed commets regardg ths burde estmate or ay other aspect of ths collecto of formato, cludg suggestos for reducg ths burde, to Washgto headquarters Servces, Drectorate for Iformato Operatos ad Reports, 5 Jefferso Davs Hghway, Sute 04, Arlgto, VA 0 430, ad to the Offce of Maagemet ad Budget, Paperwork Reducto Project ( ) Washgto DC AGENCY USE ONLY (Leave blak). REPORT DATE Jue 0 4. TITLE AND SUBTITLE Costructo ad Aalyss of Mult-Rate Parttoed Ruge-Kutta Methods 6. AUTHOR(S): Patrck R. Mugg 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgraduate School Moterey, CA SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A 3. REPORT TYPE AND DATES COVERED Master s Thess 5. FUNDING NUMBERS 8. PERFORMING ORGANIZATION REPORT NUMBER 0. SPONSORING/MONITORING AGENCY REPORT NUMBER. SUPPLEMENTARY NOTES The vews expressed ths thess are those of the author ad do ot reflect the offcal polcy or posto of the Departmet of Defese or the U.S. Govermet. IRB Protocol umber N/A. a. DISTRIBUTION / AVAILABILITY STATEMENT Approved for publc release; dstrbuto s ulmted 3. ABSTRACT (maxmum 00 words) b. DISTRIBUTION CODE A Adaptve mesh refemet (AMR) of hyperbolc systems allows us to refe the spatal grd of a tal value problem (IVP), order to obta better accuracy ad mproved effcecy of the umercal method beg used. However, the solutos obtaed are stll lmted to the local Courat-Fredrchs-Lewy (CFL) tme-step restrctos of the smallest elemet wth the spatal doma. Therefore, we look to costruct a mult-rate tme-tegrato scheme capable of solvg a IVP wth each spatal sub-doma that s cogruet wth that sub-doma s respectve tmestep sze. The prmary objectve for ths research s to costruct a mult-rate method for use wth AMR. I ths thess we wll focus o costructg a d order, mult-rate parttoed Ruge-Kutta (MPRK ) scheme, such that the o-uform mesh s costructed wth the coarse ad fe elemets at a two-to-oe rato. We wll use geeral d ad 4 th order fte dffereces (FD) methods for o-uform grds to dscretze the spatal dervatve, ad the use ths model to compare the MPRK tme-tegrator agast three explct, d order, sgle-rate tme-tegrators: Adams-Bashforth (AB ), Backward Dfferetato Formula (BDF ), ad Ruge-Kutta (RK ). 4. SUBJECT TERMS Mult-rate Method, Ruge-Kutta, Fte Dfferece, Tme-tegrato, Partal Dfferetal Equato 5. NUMBER OF PAGES PRICE CODE 7. SECURITY CLASSIFICATION OF REPORT Uclassfed 8. SECURITY CLASSIFICATION OF THIS PAGE Uclassfed 9. SECURITY CLASSIFICATION OF ABSTRACT Uclassfed 0. LIMITATION OF ABSTRACT NSN Stadard Form 98 (Rev. 89) Prescrbed by ANSI Std UU

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6 Approved for publc release; dstrbuto s ulmted CONSTRUCTION AND ANALYSIS OF MULTI-RATE PARTITIONED RUNGE-KUTTA METHODS Patrck R. Mugg Capta, Uted States Army B.S., Uted States Mltary Academy, West Pot, 003 Submtted partal fulfllmet of the requremets for the degree of MASTER OF SCIENCE IN APPLIED MATHEMATICS from the NAVAL POSTGRADUATE SCHOOL Jue 0 Author: Patrck R. Mugg Approved by: Professor Fracs Graldo Thess Advsor Assocate Professor Hog Zhou Secod Reader Approved by: Professor Carlos F. Borges Char, Departmet of Appled Mathematcs

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8 ABSTRACT Adaptve mesh refemet (AMR) of hyperbolc systems allows us to refe the spatal grd of a tal value problem (IVP), order to obta better accuracy ad mproved effcecy of the umercal method beg used. However, the solutos obtaed are stll lmted to the local Courat-Fredrchs-Lewy (CFL) tme-step restrctos of the smallest elemet wth the spatal doma. Therefore, we look to costruct a mult-rate tmetegrato scheme capable of solvg a IVP wth each spatal sub-doma that s cogruet wth that sub-doma s respectve tme-step sze. The prmary objectve for ths research s to costruct a mult-rate method for use wth AMR. I ths thess we wll focus o costructg a d order, mult-rate parttoed Ruge-Kutta (MPRK ) scheme, such that the o-uform mesh s costructed wth the coarse ad fe elemets at a two-to-oe rato. We wll use geeral d ad 4 th order fte dffereces (FD) methods for o-uform grds to dscretze the spatal dervatve, ad the use ths model to compare the MPRK tme-tegrator agast three explct, d order, sgle-rate tme-tegrators: Adams-Bashforth (AB ), Backward Dfferetato Formula (BDF ), ad Ruge-Kutta (RK ). v

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10 TABLE OF CONTENTS I. INTRODUCTION... A. OBJECTIVES OF RESEARCH...4 B. APPLICATION...5 II. FINITE DIFFERENCES...7 A. SPATIAL FINITE DIFFERENCE APPROXIMATIONS...7 B. FIRST ORDER APPROXIMATION...9 C. HIGHER ORDER APPROXIMATIONS... D. DIFFERENCE REPRESENTATION OF A PDE.... Cosstecy...4. Stablty ad Covergece Vo Neuma Stablty Aalyss...6 III. TIME-INTEGRATION METHODS... A. MULTI-STEP METHODS.... Adams Methods.... Backward Dfferetato Formulas...5 B. MULTI-STAGE METHODS...9. Explct Ruge-Kutta Methods...30 C. SINGLE-RATE RESULTS ON UNIFORM GRID Explct RK Method...37 IV. MULTI-RATE METHODS...47 A. NON-UNIFORM GRIDS...47 B. MULTI-RATE GRID DEVELOPMENT...57 C. MULTI-RATE PARTITIONED RK METHOD (MPRK )...6 D. IMPLEMENTATION, RESULTS AND ANALYSIS...65 V. SUMMARY AND CONCLUSIONS...73 APPENDIX A: TSE OF EQUATION APPENDIX B: 4TH ORDER, CENTERED DIFFERENCE STENCIL...77 APPENDIX C: ND ORDER RK PROOF...8 APPENDIX D: DERIVATION OF AB AND BDF...83 LIST OF REFERENCES...87 INITIAL DISTRIBUTION LIST...89 v

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12 LIST OF FIGURES Fgure. Rsg Thermal Bubble (Ital Tme = 0 sec.)... Fgure. Rsg Thermal Bubble (Fal Tme = 000 sec.)... Fgure 3. Uform Grd Space ad Tme...8 Fgure 4. Stablty Rego for Forward Euler w/ st Order Upwd...8 Fgure 5. Cotour Plot for Forward Euler w/ st Order Upwd...9 Fgure 6. AB Stablty Rego...4 Fgure 7. BDF Stablty Rego...8 Fgure 8. RK Stablty Rego...34 Fgure 9. Stablty Regos for AB, BDF, ad RK...35 Fgure 0. Gaussa Plot for Varous Parameters...36 Fgure. Numercal vs. Exact Soluto for RK usg d Order CFD wth x = Fgure. Numercal vs. Exact Soluto for RK usg d Order CFD wth x = Fgure 3. Estmated Error vs. Courat for RK usg d Order CFD wth x = Fgure 4. Estmated Error vs. Courat for RK usg 4 th Order CFD wth x = Fgure 5. Error vs. Courat for Varous x...43 Fgure 6. Error vs. Tme-step for Varous x...44 Fgure 7. Error vs. Tme-step for RK, AB, ad BDF...45 Fgure 8. No-uform Grd Spatal Doma...47 Fgure 9. No-uform RK Results: L Error Norms vs. Courat Value usg d Order Cetered Fte Dfferece Method...53 Fgure 0. Uform vs. No-uform RK Results...54 Fgure. Estmated Error vs. Number of Pots for Uform ad No-uform RK Results...55 Fgure. Uform vs. No-Uform RK Results usg a 4 th Order CFD Stecl...56 Fgure 3. No-coformg Grd both Space ad Tme...57 Fgure 4. Lack of Iformato Needed to Buld a 4 th Order CFD Stecl...58 Fgure 5. No-coformg Grd wth Three Sub-domas...59 Fgure 6. Iterfaces ad Buffer Regos for No-Coformg Grd...60 Fgure 7. Computg a Fast or Slow Soluto the Buffer Rego...6 Fgure 8. Butcher Tableau for RK Method ad ts Assocated Slow ad Fast RK Equvalets...64 Fgure 9. RK vs. MPRK Covergece Results usg Approxmately 00 Degrees of Freedom...66 Fgure 30. RK vs. MPRK Effcecy Results usg Approxmately 00 Degrees of Freedom...67 Fgure 3. RK vs. MPRK Effececy Results (~00 DoF)...69 Fgure 3. RK vs. MPRK Covergece Results usg Approxmately 600 DoF...70 Fgure 33. RK vs. MPRK Effcecy Results usg Approxmately 600 DoF...7 x

13 Fgure 34. RK vs. MPRK Effcecy Results usg Approxmately 600 DoF...7 Fgure 35. Uform Grd Space ad Tme...77 x

14 LIST OF TABLES Table. Dfferece Approxmatos Usg Two To Three Pots... Table. Mult-step Methods of Order p =,,3 *Implct Method...5 Table 3. Evaluatos per Step ad Trucato Error...33 Table 4. MPRK Algorthm Used to Smultaeously Solve the IVP o Both the Fast ad Slow Sub-domas...65 x

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16 LIST OF ACRONYMS, ABBREVIATIONS, AND TERMS AB P BDF P Cosstecy Explct Adams-Bashforth method of order p Backwards Dfferetato Formula of order p A FD scheme s sad to be cosstet f the dfferece betwee ts approxmato to the PDE ad the exact soluto,.e., trucato error, goes to zero the lmt, as the grd or mesh s refed. For spatal FD approxmatos, a method where oe ukow dervatve value s gve by a lear combato of kow fucto values. For FD tme-tegrators, a method where the ukow (future) fucto value s gve terms of kow fucto ad dervatve values. FD Implct Fte Dfferece For spatal FD approxmatos, a method where lear combatos of ukow dervatve values are gve terms of lear combatos of kow fucto values. For FD tme-tegrators, a method where the ukow (future) fucto value cludes a term volvg a ukow (future) dervatve value. Mult-rate (O) ODE PDE Numercal method that uses multple tme-step values smultaeously, such that each elemet wth the spatal doma uses ts cogruet tmestep sze, esurg local CFL stablty Bg-O otato Ordary Dfferetal Equato Partal Dfferetal Equato x

17 RK M Sgle-rate Stablty T.E. Ruge-Kutta method of order M Numercal method that uses a sgle tme-step value to advace the soluto tme, where the tme-step used s lmted the local CFL stablty restrctos of the smallest elemet wth the grd. Numercal methods are cosdered to be stable f the errors they produce do ot grow ubouded as the umercal algorthm s calculated whle tme creases. Trucato error, whch s the dfferece betwee the exact soluto to the partal dervatve ad ts fte dfferece represetato. xv

18 ACKNOWLEDGMENTS I would lke to frst thak God for beg the ceter of my lfe ad provdg me wth the drecto ad gudace eeded to accomplsh my work. I would also lke to thak my wfe, Starrssa, ad daughter, Serra, for ther love ad support durg my graduate studes ad for ther patece ad uderstadg as I worked to complete ths thess. I would also lke to recogze Dr. Shvasubramaa Gopalakrsha ad Dr. Mchal Kopera for ther subject matter expertse ad help my research, as well as the rest of the staff ad faculty of the Departmet of Appled Mathematcs. Much apprecato s also due to Prof. Hog Zhou, for her advce ad costructve revews to the thess. Lastly, I wat to ackowledge my thess advsor, Prof. Frak Graldo, for hs ecouragemet, expertse ad fredshp throughout my two years at the Naval Postgraduate School. Wthout hs help, ths thess would ot have bee possble. Thak you. xv

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20 I. INTRODUCTION There s a ever creasg eed to develop umercal methods capable of solvg large-scale, real-world problems. I fact, may of these problems arse the physcal, bologcal ad egeerg sceces, such that we must use umercal models, because the problems caot be solved aalytcally. I other words, we seek to fd umercal methods to approxmate the soluto to a problem, whch s ofte modeled as a partal dfferetal equato (PDE), or system of PDEs. For example, the Departmet of Appled Mathematcs of the Naval Postgraduate School cooperato wth the Naval Research Laboratory, both located Moterey, Calfora are cotually refg a modelg evromet for solvg the three dmesoal (3D) compressble Euler ad/or Naver-Stokes equatos; ths s a Nohydrostatc Ufed Model of the Atmosphere, kow as NUMA. The NUMA model s costructed usg a elemet-based Galerk framework, whch allows ether cotuous (CG) or dscotuous (DG) hgh-order Galerk methods, ad ca be used as ether a mesoscale or global model, whch exhbts excellet scalg propertes. Addtoally, the model s extremely flexble ad allows the user to easly terchage ew tmetegrators, grd ad data structures, or ew bass fuctos, thereby esurg that the model s algorthmcally flexble. There s a sgfcat terest usg the dscotuous Galerk method (DG) for solvg flud dyamcs problems; studes by [] ad [], have show that the DG method s a good choce for the costructo of future o-hydrostatc umercal weather predcto models, as t combes hgh-order accuracy of the soluto wth geometrc flexblty of o-coformg (ustructured) grds. I order to crease the scale resoluto capabltes of DG, as well as to take better advatage of avalable computg power, the use of adaptve mesh refemet (AMR) for a quadrlateral o-coformg grd, s curretly beg vestgated.

21 I Fgures ad, we see a sapshot of a rsg thermal bubble beg solved o both a adaptve ad uform mesh, usg the quadrlateral based DG method. Fgure shows the soluto at tme t = 0 sec., ad Fgure s at tme t = 000 sec. Fgure. Rsg Thermal Bubble (Ital Tme = 0 sec.) Fgure. Rsg Thermal Bubble (Fal Tme = 000 sec.)

22 To motvate the usefuless of the mult-rate approach let us frst descrbe the AMR strategy. I the fgures above, we ote that the meshes were refed every secod throughout a 000 secod smulato. For example, usg a tme-step of t = would refe the mesh at every 00 tme-steps, order to mata the desred resoluto wth the areas of the grd where the temperature varato exsts. The refed elemets were chose at the respectve tme-steps by lookg at the value of the temperature wth each elemet ad the determg whether that value was below or exceeded a predetermed threshold. The total rutme of the fully resolved soluto (Fgure, rght pael) s approxmately 87 secods compared to the AMR soluto wth a rutme of 88 secods. Ths s a speed-up of early 3.6. Eve though the AMR solutos requre less rutme, the overall umercal method s stll restrcted to the smallest tme-step, whch s determed by the smallest grd sze wth the spatal doma. Therefore, both the uform ad adaptve mesh solutos requre the same tme-step sze. Although we ca reduce the computatoal cost usg AMR, we would lke to take advatage of the ocoformg grd by smultaeously usg larger tme-steps wth the coarse regos of the spatal doma, ad smaller tme-steps wth the fe regos. Ths problem of usg a tme-step sze, commesurate wth the elemet sze, s the motvato behd my research. From the lterature we fd that the curret approaches for tacklg tmetegrato clude: fully explct, fully mplct, or a combato of the two, kow as mplct/explct (IMEX) methods. Curretly, NUMA uses the IMEX tme-tegrato method, whch crcumvets the fast ad slow propagatg waves by splttg up the physcal process of the problem beg solved, such that the fast movg compoets are hadled mplctly, whle the slow movg compoets are hadled explctly. Although the mplct solutos have o restrcto o how large of a tme-step oe may take, the overall umercal method s stll restrcted to the tme-step sze determed by the explct soluto s spatal step-sze. Therefore, we look to develop a mult-rate, tme-tegrato method that wll allow multple tme-steps to be used smultaeously wth the varous elemets of the spatal doma. 3

23 A. OBJECTIVES OF RESEARCH I ths thess, we wll focus o the costructo ad developmet of a explct, secod-order, strog stablty preservg (SSP) Ruge-Kutta (RK), mult-rate method, whch we wll use to solve the frst-order, D, advecto equato wth costat wave speed. Ths mult-rate method wll the be compared to ts equvalet sgle-rate SSP RK method, usg both accuracy ad effcecy as the two prmary metrcs. The emphass of the thess s o the tme-tegrato aspects of umercal modelg, ad for ths reaso we restrct our spatal dscretzato to smple fte dfferece (FD) methods, albet of hgh order, ad geeralzed for o-uform grd spacg. We wll show that the mult-rate approach wll reta ts formal order of accuracy, whle creasg the effcecy of the umercal model. We wll show ths usg complexty aalyss (.e., operatos cout) ad measurg total wall-clock tme for the sgle-rate versus mult-rate approaches. Ths thess s orgazed to three ma chapters, ot cludg the troducto ad summary chapters. I Chapter II we wll look at how to represet the spatal dervatve of our tal value problem (IVP), by costructg a rage of fte dfferece stecls of varous orders of accuracy. We wll the dscuss how to choose whch stecl s best applcable by lookg at the accuracy, stablty ad covergece of each spatal stecl, combato wth a forward Euler scheme tme. I Chapter III, we wll shft our focus to lookg at three dfferet tme-tegrato methods wth the mult-step ad mult-stage famles. Specfcally, we wll look at the Adams-Bashforth (AB), Backward Dfferetato Formula (BDF), ad Ruge-Kutta (RK) methods (each explct tme) for represetg our temporal dervatve. We wll solve our IVP usg each tmetegrator cojucto wth the FD spatal stecls developed Chapter II. Fally, Chapter IV we troduce the dea of a o-coformg grd ad derve a explct, secod-order, SSP RK mult-rate method. Here, we use the sgle-rate results from Chapter III order to compare agast our mult-rate method. The thess wll be cocluded by a short summary of fdgs ad a dscusso o future research. 4

24 B. APPLICATION Throughout ths thess, the goverg equato we wll use as our test case, s the hyperbolc PDE u c u t = 0, (.) whch s commoly referred to as the wave equato, wth c beg the dsplacemet from rest (or wave speed). We chose to use Equato (.) because we ca easly compute the aalytcal soluto usg the method of characterstcs, whch ca oly be used for hyperbolc problems possessg the rght umber of characterstc famles. Furthermore, the prcpal types of problems solved usg NUMA are hyperbolc PDEs. I order to smplfy our problem, we wll assume that (.) s oe dmesoal, such that u c u = u c u = 0. (.) t t x Addtoally, we ca rewrte equato (.) as + c c u = 0, (.3) t x t x sce the mxed dervatve terms, ± c, cacel each other out. If we allow t x q= c u t x, (.4) the after substtutg equato (.4) to equato (.3) we fd the frst order, D, wave equato, also kow as the advecto equato to be the followg: q q + c = 0. (.5) t x Equato (.5) represets a rght movg wave for costat wave speed, such that c > 0. Lkewse, we could have also chose to smplfy equato (.3) as a left movg wave; 5

25 however, we wll ow cosder Equato (.5) as the PDE we wsh to solve umercally. Therefore, the IVP we wsh to solve s equato (.5), alog wth the proper tal ad boudary codtos, such that the problem beg solved s well-posed. For our purposes, ad ease of computato, we mpose the followg perodc boudary codtos for x [, + ] : q(, t) = q(, t) (.6) q q (, t) = (, t), (.7) x x where these boudary codtos wll be helpful as we move from our varous sgle-rate methods to the costructo of a mult-rate parttoed Ruge-Kutta method, whch we wll see Chapter IV. Lastly, we kow from the lterature, that the aalytcal soluto of our lear ad homogeeous IVP, Equato (.5), ca be represeted by d Alembert s soluto where we kow that the rght movg wave ca be defed as q(,) x t = f ( x ct). 6

26 II. FINITE DIFFERENCES I ths chapter, we wll look at the method of fte dffereces (FD) usg our PDE from Chapter I. We wll solve the tal-value problem (IVP) usg the FD approach, where we wll chage the doma of the cotuous problem to a dscretzed form. Ths meas that the depedet varable(s) wll exst oly at the dscrete pots of the grd geerated by our fte dfferece stecl [3]. Although we kow how to solve the orgal cotuous PDE aalytcally, we wll show the beeft usg fte dffereces to estmate the soluto, ad how ths method ca provde a excellet approxmato depedg o the order of accuracy desred. The reaso for choosg a PDE, whch we kow the soluto, s to esure that our umercal method approxmates the orgal PDE well. Oce we kow that our umercal method s cosstet ad accurate, we ca the apply the method to future PDEs or systems of PDEs where we may ot ecessarly kow how to compute the aalytcal soluto, or fdg the exact soluto s practcally mpossble. I these cases, we must stead rely o umercal methods that utlze computer algorthms to approxmate the solutos [4]. It s also mportat to ote that solvg a IVP umercally wth FD, we must replace all cotuous dervatves wth fte dfferece approxmatos; however, we are ot requred to use the same stecl or dfferece quotet for each dervatve wth the problem. Istead, we are free to choose multple fte dfferece schemes for each dervatve wth the PDE. We wll see ths later the chapter; however, let s frst take a look at how to buld dfferet fte dfferece schemes for approxmatg these dervatves. A. SPATIAL FINITE DIFFERENCE APPROXIMATIONS Sce the lear, frst order, -D, wave equato has oly two frst order partal dervatves, oe space, q x, ad oe tme, q t, we oly eed to choose two FD stecls. However, we wll explore other optos for tme-tegrato methods Chapter 7

27 III, where we wll see that hgher-order FD methods are ot as practcal tme. For ow, we wll focus prmarly o the spatal dervatve q x. Whe approxmatg a dervatve usg FD, we must make some basc choces that volve how we geerate the grd, whch wll serve as the dscretzed doma for each soluto space ad tme where we wat to compute the soluto of the modfed IVP [5]. Clearly, the fer the spatal grd spacg, the more solutos we are requred to compute at each tme step, ad the better our approxmato wll be to the orgal cotuous problem; however, we must cosder the cost-beeft aalyss of hgher accuracy versus computatoal cost. It s mportat to ote that a FD stecl wth hgher accuracy s ot oly a fucto of the grd spacg, but also the umber of grd pots used to approxmate the dervatve. Fgure 3 shows a geeral grd usg uform spacg both space ad tme. Ths grd s used to approxmate the dervatves for the soluto at each grd pot ( x, t ), usg the grd pot ( x, t ) ad ts eghborg pots. Therefore, f we thk of the soluto to the PDE as qxt ( ( ), t ), the q = qx (, t). Fgure 3. Uform Grd Space ad Tme 8

28 Fte dfferece methods are smple to costruct ad are beefcal whe the geometry of the IVP s regular. Therefore, for our -D wave equato we wll curretly assume we have a uform grd wth perodc boudary codtos for our aalyss. B. FIRST ORDER APPROXIMATION Usg the grd pots Fgure 3, we are able to costruct a frst order backward FD approxmato to the cotuous spatal dervatve at ( x, t ), usg such that q = qx ( xt, ) q q q = x x. (.) We beg wth the backward-dfferece formula stead of the forward-dfferece sce t s kow advace that Equato (.) s ucodtoally stable for the IVP (stablty aalyss wll be a major topc throughout ths thess ad wll be dscussed later ths chapter, ad future chapters, for all umercal methods used). Furthermore, ths s a obvous choce for the dervatve, gve that f we take the lmt of the rght had sde as x 0, we have q q q qx (, t) qx ( xt, ) x x x = lm + O( x) = lm + O( x) x 0 x 0 whch s the very defto of the frst order dervatve of q wth respect to x at grd pot ( x, t ). Therefore, we have bult a frst order approxmato to the IVP such that the soluto q s frst order accurate ( O( x)) space for a suffcetly small but fte x [3]. I fact, usg Taylor seres expaso (TSE) of the backward-dfferece formula above wll provde a more rgorous proof that Equato (.) s deed frst order accurate. Ths ca be see Appedx A, where we show that the cotuous dervatve s equvalet to the umercal approxmato plus the trucato error (T.E.), such that the T.E. = O( x). 9

29 Note that the forward-dfferece formula q q+ q = + O( x) (.) x x s also a frst order accurate approxmato to q x. The bg O otato used above has a very accurate mathematcal meag, such that whe we replace T.E. wth O( x), ths represets the trucato error as beg bouded by a postve real costat K, multpled by the absolute value of x, where x s reasoably small, such that we ca wrte TE.. K x as x 0. I other words, the trucato error s o the order of x rased to the hghest power foud wth all the terms the trucato error [3]. We ca thk of (O) otato as the order of accuracy for the umercal method, such that f we look at all of the terms the TSE, ad keep oly the term wth the largest growth rate, the O( x) meas we are keepg all terms of x, ad throwg away all other terms ( x ) ad hgher. It s mportat to ote that O( x) does ot ecessarly form us o how large the trucato error s, yet t does provde us a dcato o how the umercal method performs as zero. Ths s accomplshed by refg our grd spacg. x gets closer ad closer to For example, f our umercal method s o the order of O ( x ), the as we decrease such that x by half, the estmated error of the method wll decrease by a factor of four, x x O = O. 4 Results for ths ca be foud Chapter III, where we wll look at varous umercal tme-tegrato methods for solvg our IVP. For more formato o (O) otato, refer to [6]. 0

30 C. HIGHER ORDER APPROXIMATIONS There are may ways we ca develop FD approxmatos for q x. The ormal coveto for buldg these represetatos begs by choosg the specfc grd pots we wat to use to approxmate the dervatve, ad the expadg each pot about the grd pot ( x, t ). The most commo frst order dervatve represetatos ca be foud Table, where we oly use two to three grd pots to represet these dervatves [3]. Dervatve Fte-Dfferece Represetato Equato q x = Table. q q + O( x) (.3) x q+ q + O( x) (.4) x q+ q + O( x ) (.5) x 3q + 4q+ q+ + O( x ) (.6) x 3q 4q + q + O( x ) (.7) x Dfferece Approxmatos Usg Two To Three Pots Equatos (.3) ad (.4) are the backward ad forward FD approxmatos respectvely, whereas Equato (.5) s the cetered-dfferece approxmato whch ca be computed by subtractg the TSE of qx ( xt, ) from the TSE of qx ( + xt, ) about the pot q. Ths gves us a secod order accurate dfferece formula for the frst dervatve usg the pots q + ad q. Appedx B outles how we ca costruct a 4th order, cetered dfferece stecl for the frst dervatve, whch wll be used cojucto wth the tme-tegrato methods developed Chapters III ad IV to aalyze sgle ad mult-rate methods both uform ad o-uform grds. Ths stecl s 4th order, such that

31 q q 8q + 8q+ q+ = + O x x 4 ( x ). (.8) It s easy to see how we ca costruct ay FD stecl for frst, secod or eve hgher order dervatves, as well as mxed partal dervatves, f desred, by usg the same approach outled Appedx B. However, the PDE we wll use oly requres frst order dervatves; therefore, we wll ot eed to costruct ay more FD stecls tha what has already bee show. D. DIFFERENCE REPRESENTATION OF A PDE From Chapter I we saw that the IVP questo s the -D, frst-order wave equato, wth costat wave speed, c, such that q q = c, c> 0. (.9) t x Usg ether equatos (.3) ad (.4), or the grd pots foud Fgure, we ca rewrte our PDE a frst order accurate dscretzed form for both space ad tme. We wll use forward Euler to represet the temporal dervatve, ad Equato (.3), frst order backward-dfferece formula, for the spatal dervatve, such that + q q q q q q = ad =. t t x x Substtutg these back to Equato (.9), we the have the followg approxmato to our PDE: q q q q = c t x + where we ca rewrte (.0) such that ( ), (.0) t, where. (.) + q = q σ q q σ = c x

32 We wll see later o the chapter, that the value of σ (Courat umber) wll play a mportat role determg where our umercal method s stable, as t measures how fast formato flows across the grd pots. Equato (.) s the explct forward Euler (frst-order upwd) represetato of the PDE (.9) usg a frst order FD stecl space. The method s cosdered explct sce we are solvg the equato for oly oe ukow value, q +, whch s the soluto at the ext tme step. As prevously stated, the spatal ad temporal FD schemes do ot have to be the same; therefore, f we fx the tme stecl to be forward Euler, we have a fte umber of optos for represetg the spatal dervatve. We could choose to use a hgher order method space; however, the overall accuracy of the combed umercal method wll oly be as good as the weakest lk. Ths meas that f we use a frst order method tme ad a fourth order method space, the overall method s stll oly frst order accurate. Below are a few more examples for the explct forward Euler method: q q q q = c t x + + (.) q q 3q 4q + q = c t x + (.3) methods s Both methods above are secod order accurate space such that the T.E. for each O t x (, ). Ths s mportat, gve that we ca show that the accuracy space ad tme are depedet of each other. However, there are may factors we must cosder whe determg whch umercal approxmato method to use whe solvg a partcular problem. Of these, accuracy, cosstecy, stablty ad effcecy must be aalyzed. Although we wat a umercal method that s as accurate as possble, cosstecy ad stablty are of much more mportace, such that f the method s both cosstet ad stable, the t wll coverge to the correct soluto [7]. However, f we have two methods that are both stable, cosstet, ad have the same order of accuracy, the we mght wat to look at whch method s most effcet (.e., lower computatoal cost). It s mportat to ote that regardless of how accurate or effcet the method s, f we do ot have both stablty ad cosstecy, the our umercal soluto wll be of o 3

33 value [8]. Therefore, we must frst esure the method s stable ad cosstet, ad the we ca look for how to crease the accuracy or effcecy of the method, depedg o what s of most mportace to us.. Cosstecy We have looked at the accuracy of a sgle FD stecl; however we have ot show how to determe the accuracy of the etre umercal method. Usg Equato (.0), we ca show the accuracy of the method by frst applyg TSE about the pot ad the substtutg to the umercal method. + q q ( t) 3 q = qx (, t+ t) = q + t+ + O( t) t t! q q q ( x) q qx xt q x O x x x! 3 = (, ) = + ( ) Substtutg these expasos to Equato (.0) gves us the followg: q q ( t) 3 q q ( x) 3 q + + t + O( t ) q ( ) q q + x O x t t! x x! = c t x where the above equato reduces to: q q t q q x + + O( t ) = c O( x ) + t t! x x! q t q + c + O( t, x) = 0. (.4) x Thus, we have prove that method (.0) s frst order accurate both space ad tme. Now, a umercal method s sad to be cosstet f the dfferece betwee ts approxmato to the PDE ad the exact soluto,.e., trucato error, goes to zero the lmt, as the grd s refed. Therefore, f we allow both the temporal grd spacg 4

34 parameter, t, ad spatal grd spacg parameter, x, Equato (.4) to both go to zero the lmt, such that ( t, x) 0, the we ca clearly see we wll recover the orgal cotuous PDE, Equato (.9), thereby showg the umercal method, (.0), s cosstet. Ths same procedure s vald for ay umercal method, where more formato o cosstecy ad stablty for a geeralzed dscretzato ca be foud most umercal textbooks.. Stablty ad Covergece Whe decdg whether or ot a FD stecl s approprate to use solvg a IVP, we have looked at the propertes of accuracy ad cosstecy; however, oe of the most crtcal propertes to aalyze, s the stablty of the method. I fact, ot oly stablty, but also the rate at whch the umercal solutos for the method beg used coverge to the true soluto of the PDE. Oe way to measure the dfferece betwee the true soluto, whch we wll defe as φ, ad our umercal soluto, q, s to take the orm of the dfferece betwee the two solutos at each grd pot [9]. Although there are may forms for computg a orm, we wll use the Eucldea, or L, orm, whch s defed as q N q. = Therefore, we wat to see that the lmt, as ( t, x) 0, the value of the orm betwee the umercal soluto ad the true soluto coverges o the order of r s O( t, x ), where r ad s represet the order of accuracy, such that φ q = O( t, x ). r s Note that ths umercal method satsfes the Lax Equvalece Theorem, whch states that f a FD scheme s lear, stable ad accurate of order (,) rs, the t s coverget of order (,) rs [9]. I other words, ths theorem establshes the fact that f a uque soluto exsts for our IVP, ad the soluto depeds cotuously o both the 5

35 tal data ad boudary codtos, the the problem s cosdered to be well-posed; such that, f we have a cosstet ad stable method, the the umercal method wll deed coverge to the true soluto of the orgal PDE [8]. The above theorem s extremely mportat ad forces us to ot smply develop a FD scheme that s accurate or cosstet. Accordg to [9], cosstecy s ot eough to assure the covergece of a umercal method. The method must also be stable. There are a great umber of cosstet fte-dfferece methods that are utterly useless because they are ustable. Therefore, f we retur to the defto of stablty, we must esure that the umercal soluto, q, does ot grow wthout boud, such that for every tme step t, less tha the fal tme T, we ca fd a costat C T that satsfes 0 q CT q, for all values of t ad x, that are reasoably small [9]. 3. Vo Neuma Stablty Aalyss We have just see the mportace of stablty; however, we would ow lke to exted ths dea to our partcular problem at had, such that usg Equato (.0) or (.), we ca fd for what values of σ our method wll be stable. Ay value that satsfes the stablty codtos wll the fall to what we call the stablty rego. The most wdely kow ad used procedure for aalyzg stablty s the Vo Neuma Method, such that Vo Neuma s stablty aalyss looks at the dscretzed soluto for a gve tme step, ad represets ths soluto as a fte Fourer seres, where the overall stablty of the umercal method ca be determed by aalyzg each elemet or Fourer mode of the seres [9]. I ths maer, we ca esure that the overall stablty of our method wll deed be stable f every Fourer mode s stable. Below s the Fourer seres represetato for the umercal soluto q j, j N l= N ( ) l q = q e l k j x 6

36 where ( ) q l represets the ampltudes of the wave, =, k l are the wave umbers ad kl x represets the phase agle, sometmes deoted φ l, whch determes the low from hgh frequecy waves, such that a phase agle of zero dcates a large ampltude wave ad a phase agle of π dcates a small ampltude wave [8]. It would be extremely tedous to evaluate all Fourer modes; therefore, f we stead look at just oe, where we esure that ths mode has a ampltude ( ) q, that does ot grow for all tme, ad s bouded such that q ( ), the we ca determe where our rego of stablty les. For more detaled formato o Vo Neuma s Method, look to [3], [8], ad [0]. If we ow cocetrate o the sgle Fourer mode ( ) j qj q e φ =, the we ca substtute ths mode to Equato (.), wth the followg statemets: q = Ge, q = G e, q = Ge, jφ jφ ( j ) φ j j j such that we have where G replaces ( ) G e = Ge σ Ge Ge + jφ jφ jφ ( j ) φ ( ) q as the ampltudes. If we ow smplfy the above expresso by dvdg through by j Ge φ, we fd that G = σ ( e φ ). ± φ Usg Euler s formula, e = cosφ± sφ, offers G = σ( cos φ) σ s φ, (.5) whch we kow s the graph of a crcle cetered at ( σ, 0) wth radus σ. Note that we wat to solve Equato (.5) for σ values that satsfy G where 7

37 c t σ x. (.6) From Equato (.5) we fd that the soluto s σ ad s referred to as the Courats-Fredrchs-Lewy (CFL) codto, whch esures that our soluto for ths specfc umercal method wll be stable wth the stablty rego defed by σ [0]. Ths ca be see Fgure 4. Ths fgure depcts the regos of stablty for Equato (.), such that stablty s mataed for values sde of the ut crcle (sold blue le), gve a partcular Courat umber. Several values for σ have bee provded, whch demostrates that for 0 σ, the method s stable. Aother method for aalyzg the stablty rego s to look at the magtude of the amplfcato factor G, such that for G the method s stable. Fgure 5 shows a cotour plot, where each cotour le represets a value of G for 0.5 σ.5 ad 0 φ π. Here we chose a few values of σ >, order to verfy that they deed correspod to values of G >, therefore provg that Equato (.) s ustable for Courat values greater tha oe. Fgure 4. Stablty Rego for Forward Euler w/ st Order Upwd 8

38 Fgure 5. Cotour Plot for Forward Euler w/ st Order Upwd 9

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40 III. TIME-INTEGRATION METHODS I Chapter II we troduced fte dfferece (FD) methods ad demostrated how we ca use these methods to approxmate spatal dervatves. FD are based o Taylor seres expaso (TSE), ad are also used to derve tme-tegrato stecls, whch we wll use to solve our tal value problem (IVP). I ths chapter, we wll tur our focus to two braches of tme-tegrato methods kow as mult-step ad mult-stage. We wll ot go to great detal for ether of these methods, as they ca be foud most umercal aalyss texts. However, we wll brefly troduce these methods for the purpose of comparg each method wth sgle-rate results usg our partcular IVP. A. MULTI-STEP METHODS Mult-step methods are all methods that requre startg values from several prevous tme steps. These methods ca be very useful; however, they requre pror formato from the system beg solved. Therefore, they must frst be started by usg a tal value, the mplemetg a oe-step method to receve two values, the a twostep method to receve three values, ad so o, depedg o the order of the method. I fact, Forward Euler (explct method) ad Backward Euler (mplct method) are frstorder mult-step methods, although they both cota oly oe-step. Other mult-step methods clude the Trapezodal ad Leapfrog schemes, whch are both mplct, secod-order accurate methods. I geeral, there are two partcular famles of mult-step methods: the Adams methods ad the Backward Dfferetato Formulas. We wll start by lookg at the Adams methods.. Adams Methods Wth the Adams famly of mult-step methods, the two most commoly used are Adams-Bashforth of order p, (AB P ), ad Adams-Moulto of order p, (AM P ), For our

41 purposes, we wat to oly focus o explct tme tegrato methods; therefore, we wll oly look at AB P, as these methods are explct tme, whereas the AM P are all mplct tme. The geeral formula for the Adams-Bashforth method for solvg the IVP, s q' = Ftq (, ), qt ( ) = q, t t, = m + m j + j + j j= q q t b Ft (, q ), (3.) such that ay AB P method ca be derved by Equato (3.), whch ca be costructed usg ether Taylor seres expaso or Lagrage terpolatg polyomals. If we tur 3 our focus to AB, where m =, b =, ad b 0 = usg Equato (3.), we ca see that we acheve the followg formula, t q + = q + ( 3 Ft (, q) Ft (, q )), (3.) whch s a two-step method, ad requres the solutos at two prevous tmes, q ad q, order to compute the ext soluto, q +, whch s at tme t +. Therefore, we ca use ay sgle-step method we lke to jumpstart AB ; however, order to mata d order accuracy, the sgle-step method must also be d order accurate. For a more detaled descrpto of how Equato (3.) s costructed usg TSE, look to Appedx D. Before usg AB wth our IVP, t s ecessary that we kow the stablty rego of ths method; therefore, f we frst assume that there s o error wth our spatal dscretzato method, the f we let Ft (, q ) = λq, ad (, Ft q ) λq =, the whe we perform the Vo Neuma stablty aalyss for AB, we have the followg:

42 where we let q z = + ( 3 ), where = λ + q q q q z t ( + ) θ θ z θ ( ) θ e e = ( 3 e e ), = e θ, because we seek to fd the curve for whch q =, ad defes the boudary of stablty. The, solvg for z, we fd: (cosθ + s θ) z =. 3 cosθ + sθ If we ow separate the real from the magary, we fd that: cos θ + cosθ + s θ Re( z) = (3 cosθ) + s θ 8 6 4sθcosθ + 8sθ Im( z) = (3 cosθ) + s θ, such that, plottg the Re( z ) versus the Im( z ) provdes the rego of stablty for AB, whch s show Fgure 6. From the fgure, we see that for ay value of z = λ t chose wth ths rego, the solutos to the IVP wll rema stable. However, as stated prevously, ths s assumg we have zero error wth our spatal dscretzato method. Ths s mportat as t allows us to see where each tme-tegrato method s stable, depedetly of the spatal scheme. 3

43 Fgure 6. AB Stablty Rego We have ow see the geeral formulas for Forward ad Backward Euler, ad AB. I fact, usg the followg mult-step formula, m m + j j j = j + j j= 0 j= q aq t bft (, q ), (3.3) f we let m =, the we have a oe-step method, such that Equato (3.3) becomes, ( (, ) (, )) q = aq + tb Ft q + bft q Now, f we let a0 = b0 =, ad b = 0, we fd that the above equato becomes q + = q + tf( t, q ), whch s the exact formula for the explct Euler Method. Therefore, we see that Euler s deed a mult-step method for m-step equal to oe, ad s equvaletly a Adams- Bashforth method of order oe (AB ). Now, usg Equato (3.3) ad the coeffcets from Table, we ca formulate a few of the most commoly used mult-step, tme-tegrato methods of order p =,, 3. 4

44 Method (P) Euler () Backward Euler * () Trapezodal * () Leapfrog () BDF * () AB () Coeffcets a0 = b0 =, b = 0 a =, b =, b = a0 =, b = b0 =, b = 0 a = 0, a =, b = b = 0, b = a0 =, a =, b =, b0 = b = a0 = a 3 = b = 0, b0 =, b = AM * (3) Table. a0 =,, a 5 8 = 0, b =, b0 =, b = Mult-step Methods of Order p =,,3 *Implct Method From Table we also otce that depedg o the value of the b j coeffcet, for j =, the Equato (3.3) wll geerate ether explct or mplct tme-tegrato methods. I other words, for all values of b 0, the the mult-step method wll be mplct tme.. Backward Dfferetato Formulas Aother famly of mult-step methods s kow as the Backward Dfferetato Formulas (BDF) of order p, such that BDF represets a two-step method, whch s secod order accurate tme. It s mportat to ote that the umber of steps the method does ot ecessarly determe the order of accuracy of the method. I geeral, t s possble to prove that the maxmal order of a coverget mplct, m-step method, usg Equato (3.3), s at most m + f m s eve, ad m + f m s odd; however, for explct schemes, a m-step method caot atta a order greater tha m []. Ths s kow as the frst Dahlqust barrer. For example, Table lsts AM as a two-step method; however, ths s a mplct tme-tegrato method wth 3 rd order accuracy tme. 5

45 Although Equato (3.3) s the geeral equato for all mult-step methods, we saw that there was a more geeral expresso for the famly of AB P methods usg Equato (3.). Ths s also true of BDF P, such that we ca express every BDF P method usg the followg equato: m j= 0 j a q + = tf( t +, q + ), (3.4) m j Prevously, we showed that AB was equvalet to Forward Euler (explct method). I a smlar fasho, we ca show that Backward Euler (mplct method) s equvalet to BDF, for m=, a0 = a =. Yet, of more terest to us, s BDF, as ths method wll provde us wth a addtoal d order tme-tegrato method to use for our IVP. However, gve that we wat to oly focus o explct tme-tegrators, we must look at how we ca rewrte BDF so that t s stead explct tme. Frst, let us cosder BDF : 4 t q + = q q + Ft ( +, q + ) (3.5) We ca see that the mplct form of BDF requres formato from two prevous tme steps, addto to the soluto at tme t +. I order to solve for q +, we would eed to solve a system of equatos; therefore, order to remove the mplctess of the equato, we ca stead use a approxmato for Fq + ( ) by the method of extrapolato. Sce we ca solve the IVP for the solutos at we say that Fq Fq, the f ( ) ad ( ) F = Fq Fq F = Fq Fq + + ( ) ( ) ad ( ) ( ), ad the approxmate + F F, we wll fd that Fq Fq Fq Fq + ( ) ( ) ( ) ( ) Fq Fq Fq + ( ) ( ) ( ). 6

46 However, ths approxmato uses the assumpto that we ca use a lear extrapolato. Therefore, f we approxmate Fq + ( ) by Fq Fq usg TSE ( ) ad ( ) about Fq + ( ), we fd: + + Fq ( ) ( a) Fq ( ) = Fq ( ) + t O( t) t + + Fq ( ) ( b) Fq ( ) = Fq ( ) + t O( t). t Now, f we multply (a) by ad subtract (b), the we get a expresso for such that, Fq ( + ), Fq Fq Fq + ( ) = ( ) ( ). Although both methods provded the same expresso for Fq + ( ), we have foud that the soluto usg the TSE approach s exact, whereas, we caot always guaratee the lear extrapolato method wll work. If we ow substtute ths approxmato back to Equato (3.5), we have a explct formula for BDF, such that + 4 t q = q q + ( Ft (, q) Ft (, q )). (3.6) Refer to Appedx D for a more thorough dervato of Equato (3.6). Let us ow look at the Vo Neuma stablty aalyss for Equato (3.6), where we aga assume there s o spatal error, such that, + 4 z q = q q + ( q q ), where we let θ ( ) θ + ( + ) θ q = e, q = e, ad q = e, such that factorg out the term e θ provdes the followg expresso: Now, solvg for z, we fd that θ 4 θ z θ e = e + ( e )

47 4 4 cosθ + sθ z =. 4 cosθ + sθ Therefore, separatg the real from the magary, we fally have the followg: cos + cos + s Re( z) = cos θ 4cosθ + 4 θ 6 θ θ 4 3sθcosθ + 4sθ Im( z) = cs o θ 4cosθ + 4, such that, a plot of the stablty rego for BDF s show Fgure 7, where we see that for ay value of z = λ t chose wth ths rego, the solutos to the IVP wll also rema stable assumg o spatal error. Fgure 7. BDF Stablty Rego 8

48 B. MULTI-STAGE METHODS Lke mult-step methods, mult-stage methods have the desrable property that they ca acheve very hgh-order accuracy, whle smultaeously reducg the amout of dervatves that eed to be computed at each grd pot. I order to desg a mult-stage method, let us frst cosder the ordary dfferetal equato (ODE), y' = Fty (, ), such that f we tegrate from tme t to t +, the we ca show that, + + t t + ( ) ( ) = '() = (, ()). yt yt y tdt Ft yt dt t t If we ow apply the trapezodal rule, we get + t + + yt ( ) yt ( ) = ( Ft (, yt ( )) + Ft (, yt ( ))) + O( t), whch s the Trapezodal method Table, such that, + t + + y = y + ( Ft (, y) + Ft (, y )), where the global error s O ( t ) ; therefore, ths method s a secod order, mplct multstep method. It s mportat to ote, that geeral, where we have wrtte t, we typcally wrte h, such that h s ay step sze, ad s ot restrcted to oly tme. However, for our purpose solvg the IVP, we wll use these methods as tme-tegrators where we let h = t. Although we have a expresso for a mplct, mult-step method, we stead wat a explct, mult-stage method. To accomplsh ths, we wll ow use Forward Euler to solve the ODE for y +, such that Euler s method wll be called a predctor, ad labeled as y +. We the substtute to the Trapezodal method, kow as the y + corrector, whch gves us the followg: + Euler: y = y + tf( t, y ) (predctor) t ( ) Trapezodal: y = y + Ft (, y) + Ft (, y ) (corrector) 9

49 The above combato of Euler ad Trapezodal methods s referred to as Heu s method, ad s ow a explct, secod order, mult-stage method. We may also choose to look at the above method by wrtg t the followg way: k = tf( t, y ) k = tf( t, y + k ), (3.7) + ( ) + y = y + k+ k where the umber of stages s represeted by k, ( =,,p). Here, Heu s method, Equato (3.7), s kow as a two-stage method, for p =.. Explct Ruge-Kutta Methods We have just show how to costruct a very smple, explct, mult-stage method, where Heu s method s classfed wth a much larger famly of mult-stage methods kow as Ruge-Kutta of order M (RK M ). Here, Heu s method s most commoly referred to as RK. way: I geeral, ay explct p-stage RK M method ca be costructed the followg k = tf( t, y ) k = tf t + d t, y + cjk j, =,..., p, j= p + y = y + bk j j j= where ay RK M method s determed by the coeffcets p, c, d, ad b, whch are typcally dsplayed a table referred to as a Butcher tableau, after J.C. Butcher [0]. 30

50 p Number of stages. Ths s ot to be cofused wth the order of the method; hece, chage of subscrpt to M RK M. For example, there are ffth order RK methods wth sx stages. c A p x p coeffcet matrx. d Row vector of sze p. b Row vector of sze p. Although we wll ot focus o the dervato of how these parameters are chose for specfc RK M methods, the reader ca fd more formato Numercal Methods for Ordary Dfferetal Equatos, by Butcher [0] or Numercal Aalyss, by Burde ad Fares []. Below are examples for RK, RK 3 ad RK 4, wth ther assocated Butcher Tableau: k = tf( t, y ) RK : k = tf( t, y + k) 0 + y = y + ( k+ k) k = tf( t, y ) 0 t k k = tf t +, y + RK3 : k3 = tf ( t + t, y + k+ k) + y = y + k k + k

51 k = tf( t, y ) t k k = tf t +, y + t k RK k = tf t + y + k4 = tf ( t + t, y + k3 ) = : 3, ( ) + y y k k k3 k p Recall from Chapter II, f our umercal method s of order Oh ( ), the as we reduce the step sze by a factor of two, the estmated error of the method s reduced by a factor of p. Therefore, for ay p-th order method, f we let p = M ad h= t, the for RK 4, whch has global trucato order of O 4 ( t ), 4 4 t t O = O 6. (3.8) I geeral, both mult-step ad mult-stage methods have ther advatages ad dsadvatages. Hgher order mult-step methods requre more use of past values, whereas mult-stage methods requre more calculatos per step. For RK methods, the ma computatoal effort s the evaluato of the rght had sde (RHS) fucto tself [0]. For example, the secod-order RK methods, the local trucato error s O ( t ), where the cost s two fuctoal evaluatos per step, gve there are two stages wth the step [0]. Lkewse, the fourth-order RK methods requre four fuctoal evaluatos per step, as there are four stages, ad the local trucato error s O 4 ( t ). However, ths patter does ot hold for all RK methods. I geeral, Table 3 shows the relatoshp betwee the umber of evaluatos per step ad the order of the local trucato error [0]. 3

52 Number of Evaluatos / Step ad Order of the Local Trucato Error Evaluatos per step Best possble local trucato error O ( t ) O 3 ( t ) O 4 ( t ) O ( t ) Table 3. Evaluatos per Step ad Trucato Error O ( t ) O 3 ( t ) Gve the result from Equato (3.8), t s easy to see why a hgher order RK method mght be preferred to a lower-order RK, f we are cocered wth accuracy; however, Table 3 shows that comparg hgher-order RK methods ( 5 ) wth lowerorder RK methods ( 4 ), that the lower-order methods may stead be preferable to hgher-order. I other words, although the hgher order methods allow for a larger tme step ad mproved local trucato error, the overall computatoal cost creases sgfcatly. Therefore, depedg o the problem beg solved, we may be satsfed wth usg a 3 rd or 4th order RK method, whch requres a slghtly smaller tme step, compared to say, a 8th order method; however, the overall umber of fuctoal evaluatos per step s reduced drastcally. There s always a tradeoff betwee choosg a method that has the best possble local trucato error, yet mmzes cost. For our aalyss, we wll focus oly o explct, secod-order methods; therefore, let us ow look at the stablty of RK, whch wll be our prmary tme-tegrato method for evaluatg the IVP Chapter I. If we rewrte Equato (3.7) the followg way: + t q = q + ( Fq ( ) + Fq ( + tfq ( ))), (3.9) the, after applyg Vo Neuma stablty aalyss to Equato (3.9), we fd the subsequet represetato for z, such that, z z e θ + + = 0. (3.0) Here, Equato (3.0) s quadratc; therefore, we must ow solve for the roots of the 33

53 fucto, ad the plot the real versus magary values of z, for 0 θ π. Fgure 8 dsplays these results, where we ow see the stablty rego for Equato (3.9) assumg o spatal error. Fgure 8. RK Stablty Rego Although these plots are useful o ther ow, t s more beefcal f we overlay each method s stablty rego, order that we may compare AB, BDF, ad RK agast each other. Fgure 9 shows the stablty plot for each of these methods, addto to frst-order Euler, where we ca easly see that RK has a sgfcatly larger rego of stablty comparso to the other three explct methods. Also, ths plot vsually supports the fact that RK s stable for much larger tme-step values tha the other three methods. 34

54 Fgure 9. Stablty Regos for AB, BDF, ad RK C. SINGLE-RATE RESULTS ON UNIFORM GRID Let us ow look at the umercal results for solvg our IVP o a uform grd, usg the mult-step ad mult-stage tme-tegrators metoed sectos A ad B. Frst, f we choose our tal codto to be a Gaussa fucto of the form: q x c ( ) = ae, ( x b) (3.) where a, b, ad c are real postve costats, such that, - a: heght of the curve s peak, - b: ceter posto of curve s peak, - c: wdth of bell curve, 35

55 the we kow the geeral graph of ths fucto s a symmetrc bell curve that has tal eds, whch fall off to postve ad egatve fty. Fgure 0 shows three Gaussa fuctos wth varyg parameters to demostrate how each parameter affects the shape of the curve. Fgure 0. Gaussa Plot for Varous Parameters For our aalyss of the IVP, we wll set the doma to be { x x }. Addtoally, we wll mpose perodc boudary codtos, such that the solutos at the rght ad left boudares are equal to each other as the -D wave equato propagates to the rght tme, for 0 t, ad t = σ x c, where the tme-step s a fucto determed by the Courat umber, wave speed ad grd spacg. For the followg results, we wll set the wave speed to be costat at c =, σ wll vary depedg o the stablty of the umercal method ad the grd spacg, x, wll be determed by the umber of grd pots used to evaluate the soluto for a partcular tme-step. 36

56 . Explct RK Method Let us frst rewrte our IVP, Equato (.9), q q = c, c> 0, t x usg Equatos (.5) ad (3.7), where (.5) s a secod-order cetered dfferece approxmato for the cotuous spatal dervatve, ad (3.7) s the RK tme-tegrato method. Here, we kow that f the, substtutg to (3.7) we fd: q q q Fq c c O x c x x t k = tf( q ) = q q x + ( ) = = + ( ), for = ( + ) t k = tf( q + k) = tf q ( q+ q ) x t t t = q q q q + x x x q q ( ) ( ) t t t q = q + ( q+ q ) q+ ( q+ q ) t ( ) q + q q x x x x t t = q ( q+ q ) + ( q+ q + q ). x x Therefore, we ow have the followg expresso for RK usg a secod-order cetered dfferece stecl space: + t t q = q ( q+ q ) + ( q+ q + q ). (3.) x x If we fx our tal parameters to be σ = 0.5 ad x = 0.0, the whe we approxmate Equato (.9) usg Equato (3.), Fgure shows a graphcal comparso betwee the exact soluto ad the umercal FD soluto at tmes t = 0.05, 0.50,.0, ad ,

57 Fgure. Numercal vs. Exact Soluto for RK usg d Order CFD wth x = 0.0 At tme t = 0, the umercal soluto (red curve) s set equal to the tal codto (blue curve), ad as tme creases, the tal codto (Gaussa fucto) begs to propagate to the rght. From Fgure we ca clearly see that after oe revoluto, where t =, the dfferece betwee the umercal soluto ad tal codto s approxmately 0.049, ad at t = 4, the dfferece s ; where these orm values are computed usg the L orm, whch was defed Chapter II as q N q, = such that the dfferece betwee the umercal soluto, q, ad exact soluto, φ, s 38

58 N φ q, for,,3,... φ q = = It s easy to see from Fgure, that the umercal soluto, defed by the parameters above, quckly loses accuracy as tme creases, such that both the dsperso ad dsspato errors are evdet. The dsperso error s a result of the umercal soluto evolvg slower tme compared to the real soluto, whle the dsspato error s the dfferece the heght of the umercal soluto to that of the real soluto. Therefore, sce t s a fucto of the Courat umber, wave speed ad grd spacg, we have a few optos to take order to determe f we ca acheve better results for solvg (.9) usg (3.): we ca crease or decrease the umber of grd pots used to evaluate the soluto (.e., make x smaller / larger), modfy the wave speed, choose a dfferet Courat umber, or a combato of the three, so log as we esure that the stablty of (3.) s mataed. Let us look at the results for oly chagg pots from 00 to 400, the x. If we chage the umber of grd x = 0.005, ad our tme-step ow becomes, t = 0.005, compared to the prevous tme-step value of t = 0.005, for a fxed Courat umber of σ = 0.5. Rug the same smulato aga wth RK for 0 t 4, we otce from Fgure that the umercal solutos are more accurate whe usg x = 0.005, as opposed to x = 0.0 for t ; however, for t >, the soluto ot oly loses accuracy, but s ustable for the gve parameters. Therefore, we see that although the tal computatos usg a fer grd provde better results for t, the ed result s worse. I fact, what we fd s that the umercal soluto s ustable for RK usg a fxed Courat value of σ = 0.5 ad a x = Addtoally, the computatoal cost also creased as we computed values at 400 pots versus oly 00 for each tme-step. Itally, ths dd ot make sese as I assumed usg a fer grd spacg ad smaller tme step would produce better results. However, what we fd s that as we refe the grd spacg smaller ad smaller, the tme step value eeded to esure stablty must also get smaller. 39

59 Fgure. Numercal vs. Exact Soluto for RK usg d Order CFD wth x = Therefore, we eed to look at the other parameters, amelyσ, ad fd a soluto that acheves the accuracy we desre, whle smultaeously reducg the computatoal cost ad esurg stablty. Furthermore, the results above show the mportace of carryg out the smulato for creasg tme. If we had stopped the smulato after oly oe or two revolutos, we would have assumed the soluto for RK usg the fer grd for the parameters chose was better, whe fact we have show the opposte. Regardless, ether soluto s desrable. Istead, let us look at the soluto to Equato (3.) by fxg the grd spacg ad oly varyg the Courat value. If we ow plot the umercal soluto usg the Courat value ad estmated error, we fd that as the Courat value decreases, the estmated error of the soluto decreases. These results are show Fgure 3, ad they cofrm Equato (3.8), whch shows that as the tme-step s reduced by a factor of two, 40

60 the estmated error of the method s reduced by a factor of p. Here, RK s a secod order method; therefore, as t s reduced by half, the estmated error should decrease by a factor of four. Although we are plottg agast the Courat value, remember that t σ = c, x therefore, there s a drect correlato betwee σ ad t must also get smaller, gve a fxed wave speed ad grd spacg. t, such that as σ gets smaller, Fgure 3. Estmated Error vs. Courat for RK usg d Order CFD wth x = 0.0 From Fgure 3, we otce that the estmated error begs to crease for Courat values of σ. What we fd, s that ths s ot due to the tme-tegrator, but rather the spatal error troduced by our approxmato of the spatal dervatve. Here, we used Equato (.5), whch s the secod-order, cetered dfferece stecl. For our purposes, we are ot as cocered wth the spatal error, ad oly wat to focus o the errors 4

61 produced from the varous tme-tegrato methods. Therefore, f we use a spatal dscretzato stecl wth hgher-order tha the order of our tme-tegrators, we wll hopefully be able to better aalyze the solutos for decreasg tme-steps or smaller Courat umbers. For that reaso, let us ow modfy the umercal method so that we stead compute the spatal dervatve usg Equato (.8), q q 8q + 8q+ q+ = + O x x x 4 ( ) whch s the 4th order, cetered dfferece stecl costructed Appedx B. If we aga look at the estmated error versus Courat value plot usg RK tme ad Equato (.8) space, the we fd the followg results, whch are show Fgure 4., Fgure 4. Estmated Error vs. Courat for RK usg 4 th Order CFD wth x = 0.0 4

62 From Fgure 4, we ow otce that our estmated error has mproved drastcally for decreasg values of σ ; however, our soluto becomes creasgly ustable for values of σ approxmately larger tha Fgure 4 clearly demostrates the usefuless of hgher-order spatal dscretzato methods whe aalyzg lower-order tme tegrators, such that we are better able to see the covergece rates of the tme-tegrator, as well as the estmated error. For example, both Fgures 3 ad 4 show the estmated error for the etre umercal method. Therefore, we otce that the spatal error domates Fgure 3, so that we are uable to see the temporal errors; whereas Fgure 4, the spatal error s ot readly see utl we reach a Courat value of approxmately 0., thereby allowg us to more accurately aalyze the temporal error for decreasgσ. If we ow vary both the grd spacg ad Courat values, the results are show Fgure 5. Here we fd, that as the grd sze decreases, the rage of σ values must also decrease order to mata stablty. Ths plot also demostrates the trade-off betwee accuracy ad effcecy, such that as x decreases, the more accurate the RK solutos are to the exact soluto; however, the umber of computatos per step must crease. Fgure 5. Error vs. Courat for Varous x 43

63 Fgure 5 s also useful helpg to better expla the prevous RK results, whch were show Fgures ad. Sce we were usg a fxed Courat value of 0.5 ad oly varyg x, we otce that the blue curve supports the results show Fgure, as we foud that although the method gradually curred both dsspato ad dsperso errors, t was stll stable. However, for the same Courat value of 0.5, we otce that the gree curve represets the results show Fgure, such that Fgure 5 supports the fact that for x = 0.005, the umercal method s ustable. Smlar to Fgure 5, t s also useful to plot the tme-step, t, versus the estmated error of the umercal method. I Fgure 6, we otce the areas alog each curve, where the slopes are approxmately equal to. It s wth these regos that for a partcular Courat value ad spatal grd sze, that the overall umercal method RK s stable ad d order accurate. The aalytcal proof that RK s secod-order accurate, regardless of the spatal dscretzato method, ca be foud Appedx C. Fgure 6. Error vs. Tme-step for Varous x 44

64 Fgure 6 s also helpful, as t clearly shows the relatoshp betwee the tmestep sze ad estmated error, as opposed to the Courat value. It s easly see that for the gve spatal grd szes, the estmated error s reduced as x decreases; however, t s at the cost of a much smaller tme-step, whch requres more computatoal tme. I ths chapter, we preseted three dfferet tme-tegrato schemes: two multstep (AB ad BDF ), ad oe mult-stage (RK ). Therefore, let us also preset the results solvg our IVP for all three methods. Fgure 7. Error vs. Tme-step for RK, AB, ad BDF From Fgure 7, we otce that each of the three tme-tegrators coverges to the actual 4 soluto at the expected rate of approxmately two as each method s O( t, x ). We also otce that RK s stable for larger tme-step szes, whch supports the stablty aalyss plots show Fgure 9. Therefore, gve the above results, we wll use RK as the tme-tegrato scheme of choce for the remader of ths thess ad developg our mult-rate method Chapter IV. 45

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66 IV. MULTI-RATE METHODS I ths chapter, we focus o the developmet of a secod-order mult-rate parttoed RK method (MPRK ), whch uses a seres of covex combatos of Euler steps [3]. To costruct ths method, we beg by troducg a o-uform grd, where we must geeralze the spatal dervatve stecls developed the prevous chapters. A. NON-UNIFORM GRIDS I Chapters II ad III, we solved our IVP usg secod ad fourth-order accurate FD spatal dscretzato methods that were aalyzed o a uform grd. However, as we beg to costruct our mult-rate tme-tegrato method, we must ow troduce a ouform mesh both space ad tme. For ow, we wll smplfy the problem to oly ouformty space. Fgure 8. No-uform Grd Spatal Doma 47

67 Usg Fgure 8, we otce that ths grd s uform tme for costat t, yet o-uform space. Here, we let every grd pot to the left of x, for all tme, be a dstace of x apart, whle all pots to the rght of x are a dstace of x apart. It s mportat to ote, that geeral, we ca arbtrarly vary the dstace betwee ay two pots wth the o-uform grd, ad that Fgure 8 s oly oe graphcal example of how we may choose to defe our grd space. Furthermore, ths fgure demostrates that f we ow wat to buld a dscrete approxmato to the spatal dervatve of our IVP, the we wll ot be able to use the spatal stecls developed Chapter II ad Appedx B, as they all rely o a sgle x. For example, Equatos (.5) ad (.8), q q q = + x x + O( x ) q q 8q + 8q+ q+ = + O x x x 4 ( ) are the secod ad fourth-order cetered dfferece stecls, respectvely, whch we used to approxmate the spatal dervatve of our IVP Chapter III. Let us ow look at how to costruct a geeral expresso for these stecls usg the grd pots at tme, t, refereced Fgure 8. We wll beg wth the secod-order cetered dfferece stecl, such that we wat to use the solutos at q, q ad q, to approxmate the frst-order dervatve of + q wth respect to x. Therefore, we ow wrte the spatal dscretzato as a weghted sum of these solutos, where, q x = α q. (4.) + + β q + γ q We wll beg wth usg TSE about q, such that, 48

68 q q ( x x ) q q x x O x x x x = + ( + ) + + ( + ) q q ( x x ) 3 q = q ( x x ) + + O ( x x ). x x If we ow substtute these expasos to Equato (4.), the we acheve the followg expresso: q x = α q + β q + γ q + q q q ( x x ) x x x + 3 = α q + ( x+ x) + + O ( x + x) + β q q q ( x x ) 3 + γ q ( x x ) + + O ( x x ), x x Sce we wat a expresso for the secod-order, cetered dfferece stecl, we requre the followg equaltes: α + β + γ = 0 α ( x x ) γ ( x x ) = + ( x+ x) ( x x ) α + γ = 0, whch ca also be wrtte as the followg matrx problem, α 0 0 β =, p m 0 ( γ ) p ( m ) 0 49

69 where we let p = ( x+ x) ad m = ( x x ). After solvg for the coeffcets usg Cramer s rule, we fd the followg expressos: ( ) α β γ m m p p =,, = = p m m p p m m p p m m p Now, f we substtute the values for the coeffcets α, β, ad γto Equato (4.), we fd a geeral expresso for the spatal dervatve, such that ( ) q m q + m p q p q = + O ( p m), (4.) x + p m m p whch depeds o the soluto at three pots wth two dstct parameters. I order to verfy that Equato (4.) s the correct geeral formula for a secodorder, cetered dfferece stecl, the f we assume that both parameters are equal (.e., uform grd), we ca easly show that whe p = x = m, the Equato (4.) smplfes to Equato (.5), q q q = + O x x x + ( ) thereby provg that Equato (4.) s a secod-order accurate stecl. Lkewse, f we wat to costruct a geeral, fourth-order cetered dfferece stecl, the followg the same methodology as show for developg Equato (4.), we ow fd that the spatal dervatve becomes, q x = α q, (4.3) + + β q + + γ q + δ q + ηq where after usg TSE about q, we acheve the followg matrx problem: 50

70 0 p p m m 0 ( ) ( ) α p p ( m ) ( m) 0 β 3 3 γ 3 3 = 0 ( p) ( p ) ( m ) ( ) m 0 δ η ( p) ( p ) ( m ) ( m) , such that, = ( x x ), = ( x x ) p + p + = ( x x ), = ( x x ). m m Note that the geeral expresso for the matrx above s k + 0 k + k k k k k + ( ) ( ) + 0 k! k! k! k! k, where k s the order of the method, such that k s eve. coeffcets: After solvg the matrx problem, we fd the followg expressos for the 5

71 α β m m p = p + m p + m p p p ( )( )( ) m m p = m + p p p m + p p ( )( )( ) ( ( ) ) + γ = δ m p p m p m m p p m m p p m p p = m m p + m p + m m ( )( )( ) m p p η = + ( + ) + ( ( ) ). 3 m m m p p m m p p m p m m p p (4.3.) (4.3.) (4.3.3) (4.3.4) (4.3.5) Therefore, f we substtute (4.3.) - (4.3.5) to (4.3), we acheve the fourth-order, cetered dfferece stecl we desred, such that q x = α q + β q + γ q + δ q + ηq + O( ). (4.4) + + m m p p Notce, f p = m, p = m, ad p = p, the we ca easly show that Equato (4.4) s equvalet to Equato (.8), such that q q 8q + 8q+ q+ = + O x x x 4 ( ) Let us retur to our IVP, where we ow solve for solutos usg the o-uform grd Fgure 8. If we use RK as our tme-tegrator ad Equato (4.) as the spatal dervatve, the we acheve the followg results, whch are show Fgure 9:. 5

72 Fgure 9. No-uform RK Results: L Error Norms vs. Courat Value usg d Order Cetered Fte Dfferece Method From Fgure 8, we otce that our doma s parttoed to two sub-domas, such that the left sub-doma represets the course grd ad the rght sub-doma correspods to the fe grd. Wth Fgure 9, we demostrate that f we fx the fe grd, x fe = 0.0, ad vary x coarse from 0.0 to 0.08, the as the wdth of xcoarse creases, the overall estmated error of the umercal method creases. We also otce that whe we set x = x, we retur to a uform grd, such that the errors are cosstet wth the coarse fe results show Chapter III, specfcally, Fgure 3. Ths makes sese, as we should ot expect to acheve better results usg the o-uform grd, compared to the uform grd, provded the hghest resoluto, x fe, wth both grds s equvalet. It s also mportat to aalyze the L error orms of both the uform ad ouform results, such that we should expect the same covergece rates. For example, we have show for the uform grd, that f our tme-tegrato method s o the order of 53

73 O ( t ), the as we reduce the step sze by a factor of two, the estmated error of the method s reduced by a factor of p, such that for RK, where p =, we kow t t O = O. 4 However, whe solvg o the o-uform grd, the tme-step s ow determed by the hghest spatal resoluto, order to esure stablty of the overall method; therefore, s a fucto of x fe. Fgure 0 dsplays the L error orms for our uform ad ouform grd results usg RK ad Equato (4.). From ths fgure, t s easly see that for the uform grds, as x decreases by half, the estmated error of the method decreases by a factor of four; lkewse, for the o-uform grds, as t x fe decreases by half, where x = x, the overall method also decreases by a factor of four. Fgure coarse fe 0 also verfes that the o-uform grd results acheve larger errors tha the uform grd, whe both methods use the same hgh resoluto. For example, the o-uform grd usg x = 0.0 ad x = 0.04, acheves better results tha the uform grd wth fe coarse x = 0.04; however, s worse tha the uform grd wth x = 0.0. Fgure 0. Uform vs. No-uform RK Results 54

74 It s also helpful to aalyze the estmated errors of the uform vs. o-uform RK results, usg the degrees of freedom of the spatal mesh. Therefore, Fgure demostrates that as the degrees of freedom of the umercal method crease, the estmated error decreases, such that the uform grd wll acheve better results tha the o-uform grd, gve a equvalet umber of pots. Ths s the prce oe must pay for usg the geometrc flexblty of o-uform grds, whch must be used to yeld more effcet solutos. Fgure. Estmated Error vs. Number of Pots for Uform ad No-uform RK Results Although Fgures 0 ad are helpful comparg the uform ad ouform meshes, we otce that Fgure 0 prmarly shows the spatal error for the overall umercal method, whch s o the order of O( t, x ). Therefore, order to better aalyze the temporal errors, we wll ow use RK combato wth Equato (4.4), the geeral fourth-order cetered dfferece stecl, to solve our IVP, such that the overall 55

75 umercal method s o the order of 4 O( t, x ), whe we set xcoarse = xfe. Fgure dsplays these results, where we are ow able to more accurately aalyze the temporal errors for RK. Fgure. Uform vs. No-Uform RK Results usg a 4 th Order CFD Stecl From Fgure, we clearly see that the uform results are equvalet to those foud Fgure 6, ad as we expected, troducg the fourth order cetered dfferece stecl space, has allowed us to vew the temporal errors more effectvely. We also otce from Fgure, that our sgle-rate RK method performs as we expected. For example, just as we foud Fgure 0, the results above show that for a gve o-uform grd, the relatve errors are greater tha the uform grd errors, whe both methods use the same hgh resoluto. Sce we have valdated that our geeral secod ad fourth-order spatal schemes, cojucto wth our sgle-rate RK method are both cosstet ad accurate, let us ow look at how to develop our mult-rate, parttoed, RK tmetegrator. 56

76 B. MULTI-RATE GRID DEVELOPMENT I the prevous secto we looked at the sgle-rate RK results o a o-uform grd, such that the o-uformty oly exsted wth the spatal doma, as was show Fgure 8. Let us ow exted ths o-coformg grd to both space ad tme, where Fgure 3 dsplays a graphcal represetato for ths o-uformty. Fgure 3. No-coformg Grd both Space ad Tme From Fgure 3, we ow otce that the o-uformty wth the temporal doma wll preset a problem costructg our fte dfferece represetato of the cotuous spatal dervatve for specfc pots. Therefore, f we mage that every grd pot to the left of pot Fgure 3 s of dmeso x, ad that every grd pot to the rght of pot has dmeso x, the t ca be easly see, that f we wat to represet the spatal dervatve of our IVP usg the geeral fourth-order cetered dfferece stecl developed Chapter IV, that we wll have o ssues costructg ths stecl for grd pots to the left of, or rght of +, assumg that the grd exteds ftely both 57

77 drectos, thereby excludg ay boudary codtos. However, we otce Fgure 4 that as tme creases, we lack kowledge of formato at every half tme-step, that s ecessary for buldg the fourth-order cetered FD stecl for grd pots ad +. These locatos are dcated by red markers Fgure 4. Fgure 4. Lack of Iformato Needed to Buld a 4 th Order CFD Stecl Let us ow troduce some termology, such that we wll commoly refer to the coarse rego wth our spatal doma as the slow rego, ad the fe rego wth the same doma as the fast rego, where the grd pot located o the boudary betwee these two regos s a terface pot. Let us also assume from ths pot forward that whe we refer to a FD stecl, that the referece stecl s the fourth-order cetered FD stecl foud Equato (4.4) wth weghted coeffcets defed by Equatos (4.3.) (4.3.5). From Fgure 4, we otce that order to represet the spatal dervatve at grd pots ad +, that we must make a approxmato for the values at each half tme-step where we do ot have ay formato. 58

78 pots Usg the formato depcted Fgure 4, f we choose to let the values at grd x, + t ad x, + t be equal to, x t ad, x t respectvely, the t ca be easly show usg TSE, that ths approach wll reduce the order of accuracy of the overall method. Lkewse, f we choose to let these same pots be equal to, + x t, + x t ad, the ths aïve approxmato wll also reduce the overall order of accuracy. However, f we kow how to compute the formato for grd pots ad at tme levels t ad t +, the a better approach would be to take a average of the two values ad use ths formato as placeholders for approxmatg our dervatve, such that we ca easly show usg TSE, that ths averagg approach wll mata the desred 4 level of accuracy, such that our FD scheme wll rema O( x ). For our partcular IVP, we are cocered wth the boudary codtos, gve that we eforce perodcty at these boudares. Therefore, order to smplfy the problems assocated wth a o-coformg grd both space ad tme at these boudares, let us defe a ew grd that cossts of three sub-domas (coarse / fe / coarse). Fgure 5 shows a graphcal represetato of ths grd, where the frst ad thrd sub-domas (slow regos) cosst of same-szed elemets, ad the secod sub-doma (fast rego) cossts of elemets half the sze of the slow regos. Fgure 5. No-coformg Grd wth Three Sub-domas Ths grd s dfferet from the two sub-doma grd Fgure 4, that we ow have two terfaces whch we wll defe as a slow/fast terface ad a fast/slow terface, ad ca be see Fgure 6. I addto, we also have what we wll defe as buffer regos. 59

79 These buffer regos exst at each terface, such that the sze of a partcular buffer rego s depedet upo the grd pots requred to costruct the FD stecl. Frst, we must costruct slow ad fast grds, whch wll be used to compute the slow ad fast umercal solutos at a gve tme level. From Fgure 6 we otce that the buffer regos are defed as the tersectos of the slow ad fast grds. Note, t s mportat to dstgush betwee the overall coordate grd ad the three sub-grds, as both slow grds domas ad, each share grd pots wth the fast grd. Notce also that the fast ad slow grds detfed Fgure 6 are ot the same as the fast ad slow sub-domas show Fgure 5. Fgure 6. Iterfaces ad Buffer Regos for No-Coformg Grd Now that we have detfed the dfferece betwee the slow ad fast subdomas, as well as the slow ad fast grds, we must also defe where our actual umercal soluto wll exst, such that order to compute the etre soluto at a gve 60

80 tme level, we requre both the fast ad slow solutos, where the dvdual soluto at ay gve grd pot, s located oe of three soluto regos: purely fast, purely slow, or at a terface. However, the fast soluto wll take care of the terface soluto. Let us refer to Fgure 7 to see what we mea. From ths fgure, we otce that the fast soluto s computed o the grd pots wth the fast sub-doma (cludg terface pots) defed Fgures 5 ad 6, whereas the slow soluto s computed usg the grd pots wth the two slow sub-domas mus the terface pots. Fgure 7. Computg a Fast or Slow Soluto the Buffer Rego For example, f we wat to compute the FD fast soluto at the frst terface pot located wth the slow/fast buffer rego, the we requre formato from the eghborg grd pots, such that f the terface pot s grd pot, the we also eed the values at grd pots,, +, ad +, such that the values at, ad wll come from the slow soluto ad the values at +, ad + wll come from the fast soluto. However, as we saw Fgure 4, the slow solutos are computed o the slow doma, such that o formato s stored at every half tme-step. Therefore, let us ow look at how to costruct the mult-rate parttoed RK (MPRK ) tme-tegrator that wll help us overcome ths ssue. 6

81 C. MULTI-RATE PARTITIONED RK METHOD (MPRK ) As we beg to costruct our MPRK scheme, we wat to esure that all of the prevous propertes, such as cosstecy, stablty ad accuracy, as dscussed Chapter III also hold for ths tme-tegrato techque. Accordg to [3], f we cosder the secod order, Ruge-Kutta method, RK, defed by the Butcher tableau as k = tf( t, q ) RK : k = tf ( t + t, q + k) 0, (4.5) + q = q + ( k+ k) the usg the followg compact otato for explct Euler steps, ε ( tq, ) = qt ( ) + tftq (, ), (4.6) as defed [3], we ca rewrte the above RK method as a lear combato of Euler steps, such that we ca guaratee that the above method wll strogly preserve stablty. Ths s kow as a strog stablty preservg (SSP) method, such that SSP tmetegrators are methods that esure a certa orm of the soluto s bouded by the same orm of the prevous tme level, where q + q. We otce that the RK scheme (4.5) above s the same as Equato (3.9), where we have already prove Appedx C that ths tme-tegrator s deed secod order accurate, ad from Chapter III, that the above RK method, cojucto wth our fourth-order FD stecl, s cosstet wth the cotuous PDE of our IVP. Furthermore, we aalyzed the stablty of ths RK method the prevous chapter usg Vo Neuma Stablty Aalyss. Therefore, f we ow substtute a lear combato of Euler steps as defed by (4.6), to the above RK method, we fd from [3]: 6

82 + () q = q + ( k+ k), q = q + tf( t, q ) = ε ( t, q ), () (*) () () () = q + ( q + k ), q = q + tf( t, q ) = ε ( t, q ), (*) + (*) = q + q, q = ( q + q ), () = ε(0, q) + ε( tq, ), where f the forward Euler method s SSP uder ts specfc CFL tme-step restrctos, the Shu ad Osher [4] showed that hgher-order methods costructed as lear combatos of forward Euler steps wll also be SSP. Now that we have rewrtte the RK method as () q + = ε(0, q) ε( tq, ) (4.7) + the tet s to use (4.7) to develop both a slow ad fast RK scheme to be used wth the slow ad fast domas of our three sub-doma grd developed the prevous secto. We look aga to the works by Costatescu ad Sadu [3], where they show how we ca exted the above RK base method to a secod-order mult-rate parttoed Ruge- Kutta (MPRK) scheme, where the MPRK scheme ca be appled to multple parttos, wth m deotg the rato betwee the tme-steps assocated wth the fast ad slow subdomas o the same tme level. For our aalyss, we cosder oly the case where our grd s refed at most oce, such that the grd cossts of at most two levels, whch s see Fgures 5-7. Addtoally, for a two level grd, oe level correspods to the fast sub-doma, whle the other level s assocated to the slow sub-doma, such that the fast doma s refed at a two-to-oe rato. Therefore, we fd that f the slow doma has elemets of legth 63 x, tha the fast doma wll have legth x. Furthermore, the rato betwee the tmesteps assocated wth the fast ad slow sub-domas s also two-to-oe, such that m =. Let us ow look at Fgure 8, where we fd the Butcher tableau for m =, such that the base method as defed by (4.5) ca be rewrtte for the slow ad fast methods,

83 where the fast ad slow methods weght coeffcets are /mb repeated m tmes. Addtoally, the slow method repeats the base methods stages m tmes wth a tme-step of t, whle the fast method must perform m steps of the base method wth a tme-step of tm. Furthermore, [3] shows how ths techque of parttog a base RK method ca be exteded from m = to arbtrary m s. Fgure 8. Butcher Tableau for RK Method ad ts Assocated Slow ad Fast RK Equvalets We have already see how we ca rewrte the base RK method usg a lear combato of Euler steps. Therefore, let us ow represet the slow ad fast methods show Table 4 [3] usg ths same approach, where the reader s ecouraged to go to [3] for proof of coservato ad accuracy of these methods. Lookg at Table 4, we otce that each le the table s part of a teratve process for computg the solutos wth each sub-doma of the grd. For example, q F ad q are the tal solutos o both the slow ad fast sub-domas, whch tally S correspod to the exact soluto at tme t. Therefore, oce we talze the soluto vectors q F ad q, we have all the formato eeded to move to the secod le the S table, ad so o. Furthermore, the sequece whch each soluto vector s computed, resolves the ssue for how to compute the average value for grd pots wth the slow doma, that are eeded whe evaluatg our fourth-order FD stecl the buffer regos. 64

84 Fast Method Slow Method Base Method (buffer rego) (slow rego) q q q F S S q = ε (, q, q ) q = ε ( t, q, q () t () () F F F S S S F S) qs = ε ( tq, S) q = ε (, q, q ) q = ε ( tq,, q ) q = ε( tq, ) (*) t () () (*) () () (*) () F F F S S S F S S S q = ( q + q ) q = q () (*) () F F F S S q = ε (, q, q ) q = ε ( t, q, q ) q = q (3) t () (3) () (3) () F F F S S S F S S S q = ε (, q, q ) q = ε ( t, q, q ) q = q q (3*) t (3) (3) (3*) (3) (3) (3*) (*) F F F S S S F S S S = ( q + q ) q = q + q + q q = ( q + q ) + () (3*) + (*) (3*) + (*) F F F S S 4 S 4 S S S S Table 4. MPRK Algorthm Used to Smultaeously Solve the IVP o Both the Fast ad Slow Sub-domas D. IMPLEMENTATION, RESULTS AND ANALYSIS The mplemetato of ths teratve tme-tegrato process s qute smple. I fact, the most dffcult step advacg each slow ad fast soluto s wth the spatal dscretzato scheme. Therefore, oce the basc fte dfferece framework has bee establshed, as was show Secto B of ths chapter, the dvdual computatos of each Euler slow or Euler fast soluto s trval. From Table 4, we otce that the base method s equvalet to Equato (4.7), ad oly requres kowledge of the solutos o the purely slow doma. However, the fast method performs four Euler-fast steps, where each step requres kowledge from the fast soluto ad the slow soluto, f solvg for grd pots wth the buffer rego. Lkewse, the slow method the buffer rego also requres formato from the slow soluto ad fast soluto. I other words, f a grd pot lves ether a purely fast or purely slow rego, the the base method wll be used for the purely slow wth oly two Euler slow steps each of sze t, ad the purely fast wth four Euler fast steps each of sze t. It s oly wth the buffer rego that the MPRK method requres ether four 65

85 Euler-fast or four Euler-slow steps, each requrg formato from both the fast ad slow solutos to compute the RHS FD stecl. Let us ow look at the results comparg the MPRK method agast the sglerate RK for solvg our D, frst order, advecto equato wth costat wave speed ad perodc boudary codtos for the smooth Gaussa fucto o the o-coformg three sub-doma grd as defed Fgure 5. Fgure 9. RK vs. MPRK Covergece Results usg Approxmately 00 Degrees of Freedom From the above fgure, we otce that both the MPRK ad RK results coverge o the order of O ( t ) as t 0. However, for step-szes t approxmately less 3 tha0, we beg to otce the spatal error from the fourth-order FD stecl. Furthermore, we also see that for a gve tme-step sze, that the sgle-rate RK method s clearly more accurate tha the MPRK for our partcular choce of spatal elemet szes x coarse = 0.0 ad x fe = 0.0. I addto, Fgure 30 shows a plot comparg the 66

86 effcecy of each method for these same parameters, such that for ths sze of problem,.e., total umber of grd pots equal to 03, we otce that the sgle-rate RK method s also more effcet. Gve these results, we ow look to crease the umber of pots the spatal doma, to determe f/whe the mult-rate method wll be more effcet. Before we do ths, let us frst look at the computatoal effcecy of each method, such that the reaso for developg a mult-rate method the frst place was to have a more effcet umercal code capable of takg larger tme-steps wth the coarse regos ad smaller tme-steps wth the fe regos of the spatal doma. As a result, the mult-rate method should deed be more computatoally effcet tha a sgle-rate code that s restrcted to takg a tme-step depedat o the smallest grd sze elemet the spatal doma. Fgure 30. RK vs. MPRK Effcecy Results usg Approxmately 00 Degrees of Freedom 67

87 Both the sgle-rate ad mult-rate methods are computed o the same grd, such that the mult-rate method solves for the purely fast ad purely slow solutos usg the same base method as the sgle-rate method, except for the purely slow solutos are capable of takg a tme-step twce as large as the soluto computed o the fast doma. Therefore, the oly dfferece computatos s the terface regos, whch are typcally small compared to the purely fast ad purely slow regos. I other words, we truly have a mult-rate method, such that the slow regos do fact use a step-sze twce as large as the fast regos. Aalyzg the effcecy of our mult-rate tme-tegrato method, shows that the speedup of ths method s equvalet to the rato of the total work doe for the sgle-rate scheme wth ts restrctve tme-step value ( t ), to the total computatoal work of the mult-rate scheme. Therefore, f we cosder that the mult-rate method uses a step-sze of t o the fast doma, cludg terface pots, wth NF + NI grd pots, ad a step-sze of t o the slow doma wth N S grd pots, the the mult-rate speedup s mnt S = mn ( + N) + ( m/ ) N F I S, (4.8) where NT = total umber of grd pots NF = umber of purely fast grd pots, = NF + NS + NI, NI = umber of terface grd pots, m = umber of fast steps, N umber of purely slow grd pots. Furthermore, we fd that sce the total umber of terface pots N m( N, N ), the for m = 4, Equato (4.8) smplfes to S = I S F NT S N + N F S. (4.9) Addtoally, takg the lmt of S as NS NT, such that NF 0, the we fd that lm S =, (4.0) NS NT NF 0 68

88 where Equato (4.0) shows that the maxmum theoretcal speedup of our MPRK soluto s equal to, regardless of the value of m. Ths makes sese as we have assumed that there s oly oe level of refemet at a two-to-oe rato. Below, Fgure 3 shows the MPRK vs. RK wallclock tme results versus tmestep sze for x coarse = 0.0 ad x fe = 0.0. Usg (4.8), we fd that the theoretcal speedup should by approxmately 4 percet for N T = 03 grd pots. However, from ths fgure, we otce that there was o computatoal speedup. I fact, the sgle-rate method was stll more effcet for the above parameters. Note, that the dfferece betwee the theoretcal speedup ad lack of computatoal speedup ( ths case specfcally) s most lkely attrbuted to other codg effceces, ad ot the MPRK scheme. Therefore, we expect to acheve close to the theoretcal speedup as we crease the Degrees of Freedom (DoF). Fgure 3. RK vs. MPRK Effececy Results (~00 DoF) 69

89 If we ow cosder x coarse = ad x fe = , the the total umber of grd pots s approxmately 600. Fgure 3 dsplays the covergece results for these ew parameters, such that we fd the sgle-rate RK method s stll more accurate for a gve tme-step sze; however, Fgure 33 shows that the MPRK method s ow more effcet for ay gve tme-step. Fgure 3. RK vs. MPRK Covergece Results usg Approxmately 600 DoF Comparg the theoretcal versus computatoal speedup for the ewly defed parameters, x coarse = ad x fe = , the usg Equato (4.9), we otce the followg results where N T =,603, N F = 5, ad N S =,596 : S NT N + N F S.996. Ths shows that for the problem above, whch has approxmately,600 grd pots, that the theoretcal speedup should be approxmately (00% speedup). Usg the data from 70

90 Fgure 33, we fd that the computatoal speedup we acheved usg the MPRK scheme s approxmately.67 (67% speedup). These results are sgfcat, ad are exactly what we hoped to acheve. I geeral, we have prove that the mult-rate method formulated Table 4, s deed more effcet tha the sgle-rate RK. Therefore, gve that both the RK ad MPRK tme-tegrato schemes are o the order of O ( t ), the prmary cosderato oe must make betwee the two methods s whether oe desres a more accurate, but less effcet approxmato, or a slghtly less accurate, yet more effcet approxmato to the IVP. I addto, let us also look at the estmated errors of each method versus the computatoal tme, such that Fgure 34 aga shows that the sglerate RK method s stll more effcet overall reachg a specfc level of accuracy, tha the mult-rate method. Fgure 33. RK vs. MPRK Effcecy Results usg Approxmately 600 DoF Although Fgure 33 demostrates that for ay gve tme-step the mult-rate method was computatoally more effcet, the results Fgure 34 were tally 7

91 dscouragg. However, t ca be easly show that the creased accuracy of the sglerate method s attrbuted to the fxed o-uform grd, whereas the mult-rate method loses accuracy due to ths same spatal mesh. I other words, the sgle-rate method should deed be more accurate as t solves the IVP everywhere o the fxed o-uform grd wth a sgle tme-step assocated wth the smallest elemet wth the spatal doma; whereas, the mult-rate method uses a tme-step twce as large wth the etre slow sub-doma. From the results above, we see that the slow sub-doma cossted of,596 grd pots out of a total,603 pots. Although the tal grd was refed udereath the bell of the Gaussa curve, the rght movg wave was rarely sde the fast rego; therefore, the umercal soluto lost accuracy as the wave moved from the fast rego to the larger slow rego. Fgure 34. RK vs. MPRK Effcecy Results usg Approxmately 600 DoF 7

92 V. SUMMARY AND CONCLUSIONS Throughout ths thess we have show how to use the method of fte dffereces, as well as a few explct tme-tegrato schemes, order to costruct a dscretzed form of our cotuous, hyperbolc tal value problem, such that we were able to successfully solve ths IVP umercally. I addto, we demostrated how to use a lear combato of strog stablty preservg, explct forward Euler steps, order to develop a d order Ruge-Kutta, mult-rate tme-tegrato scheme, where the results from Chapter IV demostrate that ths MPRK method s deed more computatoally effcet tha ts equvalet sgle-rate RK method for ay gve tme-step sze. However, there s a great deal of research that must stll be accomplshed. Therefore, I recommed the followg areas for future research: ) Icorporate a adaptve mesh refemet method cojucto wth the multrate scheme. ) Icrease the dmesoalty of the problem. 3) Vary the tal ad boudary codtos. 4) Develop other mult-rate tme-tegrato schemes usg AB or BDF. Although the oe-dmesoal effcecy results comparg tme-step szes versus wallclock tme were successful, as show Fgure 33, t would be beefcal to see what mpact corporatg recommedato () would have o mprovg the mult-rate accuracy results dsplayed Fgure 34. If the refed sub-doma wth the spatal grd, Fgure 5, were to adapt tme to stay uder the bell of the Gaussa curve, the we should expect that the MPRK method would ot oly be more effcet for a gve tmestep sze, but also acheve a partcular level of accuracy faster tha the sgle-rate RK method. The secod recommedato s to crease the dmesoalty of the problem beg solved. If we are able to mprove both the effcecy ad accuracy of our umercal soluto oly oe-dmeso, the t follows that a adaptve mult-rate tme- 73

93 tegrato scheme should also mprove the speedup hgher dmesos. I addto to ths recommedato, t would also be beefcal to corporate other tal codtos, such that we should aalyze how well the mult-rate method s capable of solvg the same test case IVP wth ether a square wave or aother Gaussa wave wth source. Addtoally, future aalyss should be doe o other PDE s or systems of PDE s. Although the st order advecto equato was a good test case PDE, the learzed shallow water equatos, Burgers equato, or eve Maxwell s equatos, each allow for a more thorough aalyss of how well the MPRK tme-tegrato scheme compares to ay other sgle-rate method. Furthermore, although t s mportat to aalyze a partcular mult-rate approach wth ts equvalet sgle-rate method, as was show ths thess, other mult-rate tme-tegrato schemes should also be developed order to compare each mult-rate approach agast aother. 74

94 APPENDIX A: TSE OF EQUATION. We beg the proof for Equato. by usg Taylor seres expaso (TSE) to expad each grd pot about the pot We have the followg TSE for q : q, such that for q q q = + O( x) x x q q ( x) q qx xt q x O x x x! 3 = (, ) = + ( ) Substtutg q back to Equato., we the get q q ( x) 3 q q x O( x ) + q x x! = x x q q ( x) x + O x = x x = + O x x x x! 3 ( )! q q ( x) ( ) O( x) q = + O( x) x Therefore, we have show that the backward dfferece formula for the frst order partal dervatve q ( x, t ) s equal to the exact soluto plus a error term o the order of O( x). x 75

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96 APPENDIX B: 4TH ORDER, CENTERED DIFFERENCE STENCIL We beg the costructo of ths FD stecl by frst decdg whch grd pots we wat to use, ad the expadg about the pot ( x, t ) usg TSE. Fgure 35. Uform Grd Space ad Tme From Fgure 35, we ca see that there are may grd pots to choose from whe buldg a ew FD method. However, for the costructo of ths 4th order, cetered dfferece stecl, we wll use the followg grd pots: ( +, ), ( +, ), (, ), (, ) q x t q x t q x t q x t The TSE for each pot are q q ( x) q ( x) q qx xt q x O x B x x! x 3! = ( +, ) = ( ) (.) 3 q q ( x) q ( x) q qx xt q x O x B x x! x 3! = ( +, ) = [( ) ] (.) 3 77