2.29 Numerical Fluid Mechanics

Transcription

1 REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton Propertes: Consstency: Truncaton error: ) ˆ ) 0 when 0 Error equaton: ) ˆ ˆ ) ˆ ) for lnear systems) Stablty: ˆ 1 Const. for lnear systems) ) 0, ˆ ˆ ) 0 p ) ˆ ) O ) for 0 ) Convergence: ˆ 1 1 p O ) PFJL Lecture 11, 1

2 Sprng 2015 Lecture 11 REVIEW Lecture 10, Cont d: Classfcaton of PDEs and eamples Error Types and Dscretzaton Propertes Fnte Dfferences based on Taylor Seres Epansons Hgher Order Accuracy Dfferences, wth Eamples Incorporate more hgher-order terms of the Taylor seres epanson than strctly needed and epress them as fnte dfferences themselves makng them functon of neghborng functon values) If these fnte-dfferences are of suffcent accuracy, ths pushes the remander to hgher order terms => ncreased order of accuracy of the FD method General appromaton: m u m Taylor Tables or Method of Undetermned Coeffcents Polynomal Fttng) Smply a more systematc way to solve for coeffcents a j s r au j PFJL Lecture 11, 2

3 FINITE DIFFERENCES Outlne for Today Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Ellptc, Parabolc and Hyperbolc PDEs) Error Types and Dscretzaton Propertes Consstency, Truncaton error, Error equaton, Stablty, Convergence Fnte Dfferences based on Taylor Seres Epansons Hgher Order Accuracy Dfferences, wth Eample Taylor Tables or Method of Undetermned Coeffcents Polynomal Fttng) Polynomal appromatons Newton s formulas Lagrange polynomal and un-equally spaced dfferences Hermte Polynomals and Compact/Pade s Dfference schemes Boundary condtons Un-Equally spaced dfferences Error Estmaton: order of convergence, dscretzaton error, Rchardson s etrapolaton, and teratve mprovements usng Roomberg s algorthm PFJL Lecture 11, 3

4 References and Readng Assgnments Chapter 23 on Numercal Dfferentaton and Chapter 18 on Interpolaton of Chapra and Canale, Numercal Methods for Engneers, 2006/2010/2014. Chapter 3 on Fnte Dfference Methods of J. H. Ferzger and M. Perc, Computatonal Methods for Flud Dynamcs. Sprnger, NY, 3 rd edton, 2002 Chapter 3 on Fnte Dfference Appromatons of H. Loma, T. H. Pullam, D.W. Zngg, Fundamentals of Computatonal Flud Dynamcs Scentfc Computaton). Sprnger, 2003 PFJL Lecture 11, 4

5 Fnte Dfferences usng Polynomal appromatons Numercal Interpolaton: Hstorcal Newton s Iteraton Formula Standard trangular famly of polynomals + Newton s Computatonal Scheme Dvded Dfferences: c =? By recurrence: 0 0 Second dvded Frst dfferences dvded dfferences Newton s formula allow easy recursve computaton of the coeffcents of a polynomal of order n that nterpolates n+1 data pont Dervatve of that polynomal can then be epressed as a functon of these n+1 data ponts n our case, unknown fct values) PFJL Lecture 11, 5

6 Fnte Dfferences usng Polynomal appromatons Equdstant Newton s Interpolaton Equdstant Samplng Dvded Dfferences wth equdstant step sze mpled Trangular Famly of Polynomals Equdstant Samplng PFJL Lecture 11, 6

7 Numercal Dfferentaton usng Newton s algorthm for equdstant samplng: 1 st Order Frst Dervatves Trangular Famly of Polynomals Equdstant Samplng f) n=1 Frst order h PFJL Lecture 11, 7

8 Numercal Dfferentaton usng Newton s algorthm for equdstant samplng: 2 nd Order Second order f) n=2 Forward Dfference Central Dfference h h Second Dervatves n=2 Forward Dfference n=3 Central Dfference PFJL Lecture 11, 8

9 Fnte Dfferences usng Polynomal appromatons Numercal Interpolaton: Lagrange Polynomals Reformulaton of Newton s polynomal) f) 1 k-3 k-2 k-1 k k+1 k+2 Dffcult to program Dffcult to estmate errors Dvsons are epensve Important for numercal ntegraton Nodal bass n FE PFJL Lecture 11, 9

10 Hermte Interpolaton Polynomals and Compact / Pade Dfference Schemes Use the values of the functon and ts dervatves) at gven ponts k For eample, for values of the functon and of ts frst dervatves at pts k n m u u ) a ) u b ) General form for mplct/eplct schemes here focusng on space) Generalzes the Lagrangan approach by usng Hermtan nterpolaton Leads to the Compact dfference schemes or Pade schemes Are mplemented by the use of effcent banded solvers Dervatves are then also unknowns s b k k k k1 k1 k m u m j r j p q a u PFJL Lecture 11, 10

11 FINITE DIFFERENCES: Hgher Order Accuracy Taylor Tables for Pade schemes u u u 1 d e au j 1 buj cu j 1)? j 1 j j 1 j j 1 2 u j j 1 j 1 j j+1 Image by MIT OpenCourseWare. PFJL Lecture 11, 11

12 FINITE DIFFERENCES: Hgher Order Accuracy Taylor Tables for Pade schemes, Cont d α φ +1 + φ φ + α = β -1 φ φ γ φ φ Sum each column startng from left and force the sums to be zero by proper choce of a, b, c, etc: a b c d e 0 a b c d e Truncaton error s sum of the frst column that does not vansh n the table, here 6 th column dvded by Δ): 4 5 u j Image by MIT OpenCourseWare. PFJL Lecture 11, 12

13 Compact / Pade Dfference Schemes: Eamples We can derve famly of compact centered appromatons for up to 6 th order usng: α φ +1 + φ φ + α = β -1 φ φ γ φ φ Scheme Truncaton error α β γ CDS-2 CDS-4 ' Pade-4 Pade-6 ' φ 3! φ 3 3! φ 5! 5 7 φ 7! Comments: Pade schemes use fewer computatonal nodes and thus are more compact than CDS Can be advantageous more banded systems!) Image by MIT OpenCourseWare. PFJL Lecture 11, 13

14 Hgher-Order Fnte Dfference Schemes Consderatons Retanng more terms n Taylor Seres or n polynomal appromatons allows to obtan FD schemes of ncreased order of accuracy However, hgher-order appromatons nvolve more nodes, hence more comple system of equatons to solve and more comple treatment of boundary condton schemes Results shown for one varable stll vald for med dervatves To appromate other terms that are not dfferentated: reacton terms, etc Values at the center node s normally all that s needed However, for strongly nonlnear terms, care s needed see later) Boundary condtons must be dscretzed PFJL Lecture 11, 14

15 Fnte Dfference Schemes: Implementaton of Boundary condtons For unque solutons, nformaton s needed at boundares Generally, one s gven ether: ) the varable: u, t) u t) Drchlet BCs) bnd bnd u ) a gradent n a specfc drecton, e.g.: = bndt) Neumann BCs), t) ) a lnear combnaton of the two quanttes Robn BCs) bnd Straghtforward cases: If value s known, nothng specal needed one doesn t solve for the BC) If dervatves are specfed, for frst-order schemes, ths s also straghtforward to treat PFJL Lecture 11, 15

16 Fnte Dfference Schemes: Implementaton of Boundary condtons, Cont d Harder cases: when hgher-order appromatons are used At and near the boundary: nodes outsde of doman would be needed Remedy: use dfferent appromatons at and near the boundary Ether, appromatons of lower order are used Or, appromatons go deeper n the nteror and are one-sded. For eample, 1 st order forward-dfference: Parabolc ft to the bnd pont and two nner ponts: bnd Cubc ft to 4 nodes 3 rd order dfference): Compact schemes, cubc ft to 4 pts: u u u u1 u2, t) 2 1 bnd u u ) u ) u ) ) u 4u 3u ) ) ) , t) u 2u 9u 18u 11u 6, t) In Open-boundary systems, boundary problem s not well posed => u for equdstant nodes Separate treatment for nflow/outflow ponts, mult-scale embedded) approach and/or generalzed nverse problem usng data n the nteror) bnd O ) for equdstant nodes 18u 9u 2u 6 u bnd, t) u for equdstant nodes PFJL Lecture 11, 16

17 Fnte-Dfferences on Non-Unform Grds: 1-D Truncaton error depends not only on grd spacng but also on the dervatves of the varable Unform error dstrbuton can not be acheved on a unform grd => non-unform grds Use smaller larger) Δ n regons where dervatves of the functon are large small) => unform dscretzaton error However, n some appromaton centered-dfferences), specfc terms cancel only when the spacng s unform Eample: Lets defne seres at : 2 3 n n 1 n f ) f ) f ' ) f '' ) f ''' )... f ) R 2! 3! n! n1 n1) Rn f ) n 1!, n ) ) ) n n f ) f ) ) f ' ) 2! f '' ) 3! f ''' )... n! f ) R n1 ) n1) Rn f ) n 1! and wrte the Taylor PFJL Lecture 11, 17

18 Non-Unform Grds Eample: 1-D Central-dfference Evaluate f) at +1 and -1, subtract results, lead to central-dfference f ) f ) f ' ) f '' ) f ''' )... f ) R 2! 3! n! 2 3 n ) n f 1) f ) f ' ) f '' ) f ''' )... f ) Rn 2! 3! n! n n 1 1 n f ) f ) f ' ) f '' ) f ''' )... R n 1 1 2! 1 1) 3! 1 1) For a non-unform mesh, the leadng truncaton term s OΔ) The more non-unform the mesh, the larger the 1 st term n truncaton error If the grd contracts/epands wth a constant factor r e : Leadng truncaton error term s : = Truncaton error r 1 r) e e f '' ) 2 r 1 e If r e s close to one, the frst-order truncaton error remans small: ths s good for handlng any types of unknown functon f) PFJL Lecture 11, 18

19 Non-Unform Grds Eample: 1-D Central-dfference What also matters s: rate of error reducton as grd s refned! Consder case where refnement s done by addng more grd ponts but keepng a constant rato of spacng geometrc progresson),.e. r 2h 2h 1 e,2h r 1 e, h For coarse grd pts to be collocated wth fne-grd pts: r e,h ) 2 = r e,2h The rato of the two truncaton errors at a common pont s then: R 1 r ) 2h e,2h 2 1 ) h re, h f 2 f '' ) '' ) whch s R 1 r ) snce The factor R = 4 f r e = 1 unform grd). R s actually mnmum at r e = 1. When r e > 1 ependng grd) or r e < 1 contractng grd), the factor R > 4 r eh, eh, 2 r 1) 2h 1 e, h 1 PFJL Lecture 11, 19

20 Non-Unform Grds Eample: 1-D Central-dfference Conclusons When a non-unform geometrc progresson grd s refned, error due to the 1 st order term decreases faster than that of 2 nd order term! Snce r e,h ) 2 = r e,2h, we have r e,h 1 as the grd s refned. Hence, convergence becomes asymptotcally 2 nd order 1 st order term cancels) Non-unform grds are thus useful, f one can reduce Δ n regons where dervatves of the unknown soluton are large Automated means of adaptng the grd to the soluton as t evolves) However, automated grd adaptaton schemes are more challengng n hgher dmensons and for multvarate e.g. physcs-bology-acoustcs) or multscale problems Adaptve) Grd generaton stll an area of actve research n CFD Conclusons also vald for hgher dmensons and for other methods fnte elements, etc) PFJL Lecture 11, 20

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