2.29 Numerical Fluid Mechanics Spring 2015 Lecture 10

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1 REVIEW Lectre 9: Nmercal Fld Mechancs Srng 015 Lectre 10 End of (Lnear Algebrac Systems Gradent Methods Krylo Sbsace Methods Precondtonng of A=b FINITE DIFFERENCES Classfcaton of Partal Dfferental Eqatons (PDEs and eamles wth fnte dfference dscretzatons Parabolc PDEs Elltc PDEs Hyerbolc PDEs Nmercal Fld Mechancs PFJL Lectre 10, 1

2 FINITE DIFFERENCES - Otlne Classfcaton of Partal Dfferental Eqatons (PDEs and eamles wth fnte dfference dscretzatons Parabolc PDEs, Elltc PDEs and Hyerbolc PDEs Error Tyes and Dscretzaton Proertes Consstency, Trncaton error, Error eqaton, Stablty, Conergence Fnte Dfferences based on Taylor Seres Eansons Hgher Order Accracy Dfferences, wth Eamle Taylor Tables or Method of Undetermned Coeffcents Polynomal aromatons Newton s formlas Lagrange olynomal and n-eqally saced dfferences Hermte Polynomals and Comact/Pade s Dfference schemes Eqally saced dfferences Rchardson etraolaton (or nformly redced sacng Iterate mroements sng Roomberg s algorthm Nmercal Fld Mechancs PFJL Lectre 10,

3 References and Readng Assgnments Chater 3 on Nmercal Dfferentaton and Chater 18 on Interolaton of Chara and Canale, Nmercal Methods for Engneers, 006/010/014. Chater 3 on Fnte Dfference Methods of J. H. Ferzger and M. Perc, Comtatonal Methods for Fld Dynamcs. Srnger, NY, 3 rd edton, 00 Chater 3 on Fnte Dfference Aromatons of H. Loma, T. H. Pllam, D.W. Zngg, Fndamentals of Comtatonal Fld Dynamcs (Scentfc Comtaton. Srnger, 003 Nmercal Fld Mechancs PFJL Lectre 10, 3

4 Eamles: Partal Dfferental Eqatons Hyerbolc PDE: B - 4 A C > 0 (1 c t ( c 0 t (3 ( U g t (4 ( U g Wae eqaton, nd order Sommerfeld Wae/radaton eqaton, 1 st order Unsteady (lnearzed nscd conecton (Wae eqaton frst order Steady (lnearzed nscd conecton Allows non-smooth soltons Informaton traels along characterstcs, e.g.: For (3 aboe: For (4, along streamlnes: Doman of deendence of (,T = characterstc ath e.g., for (3, t s: c (t for 0< t < T Fnte Dfferences, Fnte Volmes and Fnte Elements Uwnd schemes d dt c U ( ( t c d c ds U Nmercal Fld Mechancs t 0, y PFJL Lectre 10, 4

5 Waes on a Strng Partal Dfferental Eqatons Hyerbolc PDE - Eamle (, t (, t c 0 L, 0 t t Intal Condtons t Bondary Condtons (0,t (L,t Wae Soltons (,0, t (,0 Tycally Intal Vale Problems n Tme, Bondary Vale Problems n Sace Tme-Marchng Soltons: Imlct schemes generally stable Elct sometmes stable nder certan condtons Nmercal Fld Mechancs PFJL Lectre 10, 5

6 Wae Eqaton Partal Dfferental Eqatons Hyerbolc PDE - Eamle (, t (, t c 0 L, 0 t t Dscretzaton: t Fnte Dfference Reresentatons (centered (0,t (L,t +1-1 Fnte Dfference Reresentatons (,0, t (,0 Nmercal Fld Mechancs PFJL Lectre 10, 6

7 Partal Dfferental Eqatons Hyerbolc PDE - Eamle Introdce Dmensonless Wae Seed t Elct Fnte Dfference Scheme (0,t (L,t +1-1 Stablty Reqrement: ct C 1 Corant-Fredrchs-Lewy condton (CFL condton (,0, t (,0 Physcal wae seed mst be smaller than the largest nmercal wae seed, or, Tme-ste mst be less than the tme for the wae to trael to adacent grd onts: Nmercal Fld Mechancs PFJL Lectre 10, 7 c t or t c

8 Error Tyes and Dscretzaton Proertes: Consstency Consder the dfferental eqaton ( symbolc oerator ( 0 and ts dscretzaton for any gen dfference scheme ˆ ( ˆ 0 Consstency (Proerty of the dscretzaton The dscretzaton of a PDE shold asymtote to the PDE tself as the mesh-sze/tme-ste goes to zero,.e for all smooth fnctons : ( ˆ ( ( 0 when 0 (the trncaton error anshes as mesh-sze/tme-ste goes to zero Nmercal Fld Mechancs PFJL Lectre 10, 8 (

9 Error Tyes and Dscretzaton Proertes: Trncaton error and Error eqaton Trncaton error ( ˆ ( Remember: does not satsfy the FD eqn. Snce ( 0, the trncaton error s the reslt of nsertng the eact solton n the dfference scheme If the FD scheme s consstent: ( ˆ ( ( O( for 0 (>0 s the order of accracy for the FD scheme ˆ ˆ Order ndcates how fast the error s redced when the grd s refned Error eolton eqaton From ˆ ( ˆ 0 and ˆ where ε s the dscretzaton error, for lnear roblems, we hae: ( ˆ ( ˆ ˆ ( ˆ ( The trncaton error acts as a sorce for the dscretzaton error, whch s conected, dffsed, eoled, etc., by the oerator ˆ Nmercal Fld Mechancs PFJL Lectre 10, 9

10 Stablty Error Tyes and Dscretzaton Proertes: Stablty A nmercal solton scheme s sad to be stable f t does not amlfy errors ε that aear n the corse of the nmercal solton rocess For lnear(-zed roblems, snce ˆ, stablty mles: ( ˆ 1 Const. wth the Const. not a fncton of If nerse was not bonded, dscretzaton errors ε wold ncrease wth teratons In ractce, nfnte norm ˆ 1 Const. s often sed. Howeer, dffclt to assess stablty n real cases de to bondary condtons and non-lneartes It s common to nestgate stablty for lnear roblems, wth constant coeffcents and wthot bondary condtons A wdely sed aroach: on Nemann s method (see lectres 1-13 Nmercal Fld Mechancs PFJL Lectre 10, 10

11 Error Tyes and Dscretzaton Proertes: Conergence Conergence A nmercal scheme s sad to be conergent f the solton of the dscretzed eqatons tend to the eact solton of the (PDE as the grdsacng and tme-ste go to zero Error eqaton for lnear(-zed systems: ˆ 1 ( ( Error bonds for lnear systems: ˆ1 1 ˆ11 ( Hence ˆ 1 1 O( ( ( For a consstent scheme: O( for 0 ˆ Conergence <= Stablty + Consstency (for lnear systems = La Eqalence Theorem (for lnear systems For nonlnear eqatons, nmercal eerments are often sed e.g., terate or aromate tre solton wth comtaton on sccessely fner grds, and comte resltng dscretzaton errors and order of conergence Nmercal Fld Mechancs PFJL Lectre 10, 11

12 Fnte Dfferences - Bascs Fnte Dfference Aromaton dea drectly borrowed from the defnton of a derate. 1 '( lm 0 Geometrcal Interretaton ( ( Qalty of aromaton mroes as stencl onts get closer to Central dfference wold be eact f was a second order olynomal and onts were eqally saced φ Eact Backward Central Forward Image by MIT OenCorseWare. On the defnton of a derate and ts aromatons Nmercal Fld Mechancs PFJL Lectre 10, 1

13 FINITE DIFFERENCES: Taylor Seres, Hgher Order Accracy How to obtan dfferentaton formlas of arbtrary hgh accracy? 1 Frst aroach: Use Taylor seres, kee more hgher-order terms than strctly needed and eress these hgher-order terms as fnte-dfferences themseles 3 n n 1 n f ( f ( f '( f ''( f '''(... f ( R! 3! n! n1 ( n1 Rn f ( n 1! For eamle, how can we dere the forward fnte-dfference estmate of the frst derate at wth second order accracy? 3 f ( 1 f ( f ( 1 f ( f '( f ''( O( f '( f ''( O(!! If we retan the second-derate, and estmate t wth frst-order accracy, the order of accracy for the estmate of f ( wll be = Nmercal Fld Mechancs PFJL Lectre 10, 13

14 FINITE DIFFERENCES: Taylor Seres, Hgher Order Accracy Cont d Stll sng 3 n n 1 n f ( f ( f '( f ''( f '''(... f ( R! 3! n! n1 ( n1 Rn f ( n 1! Estmate the second-derate wth forward fnte-dfferences at frstorder accracy: 3 f ( 1 f ( f '( f ''( O(! 4 3 f ( f ( f '( f ''( O(! * ( f ( f ( 1 f ( f ''( O( * (1 f ( 1 f ( f '( f ''( O(! f ( 1 f ( f ( f ( 1 f ( f ( 4 f ( 1 3 f ( f '( O( O(! Nmercal Fld Mechancs PFJL Lectre 10, 14

15 Fgre 3.1 Chara and Canale Forward Dfferences Nmercal Fld Mechancs PFJL Lectre 10, 15

16 Backward Dfferences Nmercal Fld Mechancs PFJL Lectre 10, 16

17 Centered Dfferences Nmercal Fld Mechancs PFJL Lectre 10, 17

18 Problem: Estmate 1 st derate of f = -0.1*^4-0.15*^3-0.5*^-0.5* +1. at =0.5, wth a grd cell sze of h=0.5 and sng sccessely hgher order schemes. How does the solton mroe? %Defne the fncton f=@( -0.1*^4-0.15*^3-0.5*^-0.5* +1.; %Defne Ste sze h=0.5; %Set ont at whch to ealate the derate = 0.5; FINITE DIFFERENCES Taylor Seres, Hgher Order Accracy: EXAMPLE %% Usng forward dfference %Frst order: df=(f(+h-f( / h; frntf('\n\n Frst order Forward dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 %Second order: df=(-f(+*h+4*f(+h-3*f( / (*h; frntf('second order Forward dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 %% Backwards dfference %Frst order: df=(-f(-h+f( / (h; frntf('frst order Backwards dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 %Second order: df=(f(-*h-4*f(-h+3*f( / (*h; L11_FD.m frntf('second order Backwards dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 Nmercal Fld Mechancs %% Central dfference %Second order: df=(f(+h-f(-h / (*h; frntf('second order Central dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 %Forth order: df=(-f(+*h+8*f(+h-8*f(-h+f(-*h / (1*h; frntf('forth order Central dfference: %g, wth error:%g%% \n',df,abs(100*(df+0.915/0.915 Ott Frst order Forward dfference: , wth error:6.5411% Second order Forward dfference: , wth error:5.819% Frst order Backwards dfference: , wth error:1.7466% Second order Backwards dfference: , wth error:3.7671% Second order Central dfference: , wth error:.3976% Forth order Central dfference: , wth error:.43337e-14% Why s the 4 th order eact? PFJL Lectre 10, 18

19 1 st Aroach: FINITE DIFFERENCES: Taylor Seres, Hgher Order Accracy Smmary Incororate more hgher-order terms of the Taylor seres eanson than strctly needed and eress them as fnte dfferences themseles e.g. for fnte dfference of m th derate at order of accracy, eress the m+1 th, m+ th, m+-1 th derates at an order of accracy -1,..,, 1. m General aromaton: s m a r Can be sed for forward, backward, skewed or central dfferences Can be comter atomated Indeendent of coordnate system and etends to mlt-dmensonal fnte dfferences (each coordnate s often treated searately Remember: order of aromaton ndcates how fast the error s redced when the grd s refned (not necessarly the magntde of the error Nmercal Fld Mechancs PFJL Lectre 10, 19

20 FINITE DIFFERENCES: Interolaton Formlas for Hgher Order Accracy nd aroach: Generalze Taylor seres sng nterolaton formlas Ft the nknown fncton solton of the (PDE to an nterolaton cre and dfferentate the resltng cre. For eamle: Ft a arabola to f data at onts 1,, 1 ( 1, then dfferentate to obtan: f ( 1 f ( 1 1 f ( 1 f '( ( ( ( ( ( 1 1 Ths s a nd order aromaton (arabola aro. s of order 3 For nform sacng, redces to centered dfference seen before In general, aromaton of frst derate has a trncaton error of the order of the olynomal (here All tyes of olynomals or nmercal dfferentaton methods can be sed to dere sch nterolatons formlas Polynomal fttng, Method of ndetermned coeffcents, Newton s nterolatng olynomals, Lagrangan and Hermte Polynomals, etc Nmercal Fld Mechancs PFJL Lectre 10, 0

21 FINITE DIFFERENCES Hgher Order Accracy: Taylor Tables or Method of Undetermned Coeffcents Taylor Tables: Conenent way of formng lnear combnatons of Taylor Seres on a term-by-term bass What we are lookng for, n1 st colmn: Taylor seres at: Sm each colmn startng from left, force the sms to zero and so choose a, b, c, etc Nmercal Fld Mechancs PFJL Lectre 10, 1

22 FINITE DIFFERENCES Hgher Order Accracy: Taylor Tables Cont d Sm each colmn startng from left and force the sms to be zero by roer choce of a, b, c, etc: a b 0 a b c c = Famlar 3-ont central dfference Trncaton error s frst colmn n the table that does not ansh, here ffth colmn of table: 4 4 a c Nmercal Fld Mechancs PFJL Lectre 10,

23 FINITE DIFFERENCES Hgher Order Accracy: Taylor Tables Cont d a a b / (as before and a a Nmercal Fld Mechancs PFJL Lectre 10, 3

24 Integral Conseraton Law for a scalar d dt d dv dt dv (. n da fed q.n n da s dv CV fed CM CV CV Adecte fles Other transorts (dffson, etc (Ad.& dff. fles conecton Sm of sorces and snks terms (reactons, etc CV, fed ρ,φ s Φ q Alyng the Gass Theorem, for any arbtrary CV ges:.(. q s t For a common dffse fl model (Fck s law, Forer s law: q k Conserate form of the PDE.(.( k s t Nmercal Fld Mechancs PFJL Lectre 8 N-S, 4

25 Strong-Conserate form of the Naer-Stokes Eqatons ( Nmercal Fld Mechancs PFJL Lectre 8-NS, 5 CV, fed s Φ q q ρ,φ g t g g Alyng the Gass Theorem ges: Eqatons are sad to be n strong conserate form f all terms hae the form of the dergence of a ector or a tensor. For the th Cartesan comonent, n the general Newtonan fld case: ( (... CV CV F CV d dv n da gdv dt g dv F gdv dv dv (. n da (. (. n da (. dv dv dv dv (. n da (. (. n da (. (. n da (. (. n da (. (.. n da (.. (.. n da (.. (.. n da (.. (.. n da (.. (.. n da (.. (.. n da (.. (.. n da (.. (.. n da ( CV F gdv CV CV CV CV C C C C C C C C C C F gdv F g dv g dv Wth Newtonan fld + ncomressble + constant μ: For any arbtrary CV ges:.(. 0 g t g 0 t Momentm: Mass: 3 t ge ge Wth Newtonan fld only: Cons. of Momentm: Cachy Mom. Eqn.

26 MIT OenCorseWare htt://ocw.mt.ed Nmercal Fld Mechancs Srng 015 For nformaton abot ctng these materals or or Terms of Use, st: htt://ocw.mt.ed/terms.

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12 REVIEW Lecture 11: 2.29 Numercal Flud Mechancs Fall 2011 Lecture 12 End of (Lnear) Algebrac Systems Gradent Methods Krylov Subspace Methods Precondtonng of Ax=b FINITE DIFFERENCES Classfcaton of Partal