2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12

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1 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 Dfferental Equatons (PDEs) and examples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs 2.29 Numercal Flud Mechancs PFJL Lecture 12, 1 1

2 FINITE DIFFERENCES - Outlne Classfcaton of Partal Dfferental Equatons (PDEs) and examples wth fnte dfference dscretzatons Parabolc PDEs, Ellptc PDEs and Hyperbolc PDEs Error Types and Dscretzaton Propertes Consstency, Truncaton error, Error equaton, Stablty, Convergence Fnte Dfferences based on Taylor Seres Expansons Hgher Order Accuracy Dfferences, wth Example Taylor Tables or Method of Undetermned Coeffcents Polynomal approxmatons Newton s formulas Lagrange polynomal and un-equally spaced dfferences Hermte Polynomals and Compact/Pade s Dfference schemes Equally spaced dfferences Rchardson extrapolaton (or unformly reduced spacng) Iteratve mprovements usng Roomberg s algorthm 2.29 Numercal Flud Mechancs PFJL Lecture 12, 2 2

3 References and Readng Assgnments Part 8 (PT 8.1-2), Chapter 23 on Numercal Dfferentaton and Chapter 18 on Interpolaton of Chapra and Canale, Numercal Methods for Engneers, 2010/2006. 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 Approxmatons of H. Lomax, T. H. Pullam, D.W. Zngg, Fundamentals of Computatonal Flud Dynamcs (Scentfc Computaton). Sprnger, Numercal Flud Mechancs PFJL Lecture 12, 3 3

4 Classfcaton of Partal Dfferental Equatons (2D case, 2 nd order) Quas-lnear PDE for ( x, y) y n (x,y 2 ) A,B and C Constants n (x 1,y) n (x 2,y) Hyperbolc Parabolc Ellptc (Only vald for two ndependent varables: x,y) In general: A, B and C are functon of: x, y,,, x y n (x,y 1 ) Equatons may change of type from pont to pont f A, B and C vary wth x, y,. etc Naver-Stokes, ncomp., const. vscosty: Du u 1 (u Dt t ) u p 2 u g 2.29 Numercal Flud Mechancs PFJL Lecture 12, 4 4 x

5 Partal Dfferental Equatons ELLIPTIC: B 2-4 A C < 0 y n (x,y 2 ) Quas-lnear PDE for ( x, y) A,B and C Constants n (x 1,y) D(x,y) n (x 2,y) Hyperbolc Parabolc Ellptc n (x,y 1 ) x 2.29 Numercal Flud Mechancs PFJL Lecture 12, 5 5

6 Partal Dfferental Equatons Ellptc PDE Laplace Operator Examples: Laplace Equaton Potental Flow Posson Equaton Potental Flow wth sources Heat flow n plate U u 2 u Helmholtz equaton Vbraton of plates Convecton-Dffuson Smooth solutons ( dffuson effect ) Very often, steady state problems Doman of dependence of u s the full doman D(x,y) => global solutons Fnte dffer./volumes/elements, boundary ntegral methods (Panel methods) 2.29 Numercal Flud Mechancs PFJL Lecture 12, 6 6

7 Partal Dfferental Equatons Ellptc PDEs y Equdstant Samplng u(x,b) = f 2 (x) Dscretzaton Fnte Dfferences u 0,y)=g 1 (y) u a,y)=g 2 (y) j+1 j j u(x,0) = f 1 (x) x 2.29 Numercal Flud Mechancs PFJL Lecture 12, 7 7

8 Partal Dfferental Equatons Ellptc PDE Dscretzed Laplace Equaton y Fnte Dfference Scheme u(x,b) = f 2 (x) Boundary Condtons u 0,y)=g 1 (y) u a,y)=g 2 (y) j+1 j j-1 Global Soluton Requred u(x,0) = f 1 (x) x 2.29 Numercal Flud Mechancs PFJL Lecture 12, 8 8

9 Ellptc PDE: Posson Equaton SOR Iteratve Scheme, wth Jacob 2.29 Numercal Flud Mechancs PFJL Lecture 12, 9 9

10 Partal Dfferental Equatons Hyperbolc PDE: B 2-4 A C > 0 Examples: 2 u 2 2 u (1) c t 2 2 Wave equaton, 2 nd order u u (2) c 0 Sommerfeld Wave/radaton equaton, t 1 st order u (3) (U ) u g t (4) (U ) u g Unsteady (lnearzed) nvscd convecton (Wave equaton frst order) Steady (lnearzed) nvscd convecton Allows non-smooth solutons Informaton travels along characterstcs, e.g.: For (3) above: d x c ( ( )) dt Uxc t t d x c For (4), along streamlnes: U ds Doman of dependence of u(x,t) = characterstc path e.g., for (3), t s: x c (t) for 0< t < T Fnte Dfferences, Fnte Volumes and Fnte Elements 0 x, y 2.29 Numercal Flud Mechancs PFJL Lecture 11, 10 10

11 Partal Dfferental Equatons Hyperbolc PDE Waves on a Strng uxt (, ) 2 uxt (, ) c 0 x L, 0 t 2 2 t 2 2 t Intal Condtons Boundary Condtons u 0,t) u L,t) Wave Solutons u(x,0), u t (x,0) x Typcally Intal Value Problems n Tme, Boundary Value Problems n Space Tme-Marchng Solutons: Explct Schemes Generally Stable 2.29 Numercal Flud Mechancs PFJL Lecture 11, 11 11

12 Wave Equaton Partal Dfferental Equatons Hyperbolc PDE 2 uxt (, ) 2 2 uxt (, ) c 0 x L, 0 t t 2 2 Dscretzaton: t Fnte Dfference Representatons u 0,t) u L,t) j+1 j j-1 Fnte Dfference Representatons u(x,0), u t (x,0) x 2.29 Numercal Flud Mechancs PFJL Lecture 12, 12 12

13 Partal Dfferental Equatons Hyperbolc PDE Introduce Dmensonless Wave Speed t Explct Fnte Dfference Scheme u 0,t) u L,t) j+1 j j-1 Stablty Requrement: c t C 1 Courant-Fredrchs-Lewy condton (CFL condton) u(x,0), u t (x,0) Physcal wave speed must be smaller than the largest numercal wave speed, or, Tme-step must be less than the tme for the wave to travel to adjacent grd ponts: c or t t 2.29 Numercal Flud Mechancs PFJL Lecture 11, c x

14 Error Types and Dscretzaton Propertes: Consstency Consder the dfferental equaton ( L symbolc operator) L ( ) 0 and ts dscretzaton for any gven dfference scheme Lˆ ( ˆ ) 0 Consstency (Property of the dscretzaton) The dscretzaton of a PDE should asymptote to the PDE tself as the mesh-sze/tme-step goes to zero,.e for all smooth functons : L ( ) Lˆ ( ) 0 when 0 (the truncaton error vanshes as mesh-sze/tme-step goes to zero) 2.29 Numercal Flud Mechancs PFJL Lecture 12, 14 14

15 Error Types and Dscretzaton Propertes: Truncaton error and Error equaton Truncaton error L ( ) Lˆ ( ) Remember: does not satsfy the FD eqn. Snce L ( ) 0, the truncaton error s the result of nsertng the exact soluton n the dfference scheme If the FD scheme s consstent: ( ) L ( ) O x p L ˆ ( ) for 0 p (>0) s the order of accuracy for the FD scheme Lˆ Order p ndcates how fast the error s reduced when the grd s refned Error evoluton equaton From Lˆ ( ˆ ) 0 and ˆ where ε s the dscretzaton error, for lnear problems, we have: L ( ) L ˆ ( ˆ ) L ˆ ( ) Lˆ ( ) The truncaton error acts as a source for the dscretzaton error, whch s ˆ convected and dffused by the operator L 2.29 Numercal Flud Mechancs PFJL Lecture 12, 15 15

16 Stablty Error Types and Dscretzaton Propertes: Stablty A numercal soluton scheme s sad to be stable f t does not amplfy errors that appear n the course of the numercal soluton process For lnear(-zed) problems, snce Lˆ ( ), stablty mples: L ˆ Const. wth the Const. not a functon of 1 x If nverse was not bounded, dscretzaton errors ε would ncrease wth teratons Lˆ 1 In practce, nfnte norm Const. s often used. However, dffcult to assess stablty n real cases due to boundary condtons and non-lneartes It s common to nvestgate stablty for lnear problems, wth constant coeffcents and wthout boundary condtons A wdely used approach: von Neumann s method (see lectures 13-14) 2.29 Numercal Flud Mechancs PFJL Lecture 12, 16 16

17 Convergence Error Types and Dscretzaton Propertes: Convergence A numercal scheme s sad to be convergent f the soluton of the dscretzed equatons tend to the exact soluton of the (P)DE as the grdspacng and tme-step go to zero ˆ 1 Error equaton for lnear(-zed) systems: L ( ) Error bounds for lnear systems: ˆ 1 ˆ 1 L ( ) L For a consstent scheme: O( x p ) for 0 Lˆ 1 Hence O( x p ) x Convergence <= Stablty + Consstency (for lnear systems) = Lax Equvalence Theorem (for lnear systems) For nonlnear equatons, numercal experments are often used e.g., terate or approxmate true soluton wth computaton on successvely fner grds, and compute resultng dscretzaton errors and order of convergence 2.29 Numercal Flud Mechancs PFJL Lecture 12, 17 17

18 Fnte Dfferences - Bascs Fnte Dfference Approxmaton dea drectly borrowed from the defnton of a dervatve. ( x 1 ) ( x ) '(x ) lm 0 x Geometrcal Interpretaton Qualty of approxmaton mproves as stencl ponts get closer to x Central dfference would be exact f was a second order polynomal and ponts were equally spaced φ x x x Exact Backward Central Forward Image by MIT OpenCourseWare Numercal Flud Mechancs PFJL Lecture 12, 18 18

19 FINITE DIFFERENCES: Taylor Seres, Hgher Order Accuracy How to obtan dfferentaton formulas of arbtrary hgh accuracy? 1) Frst approach: Use Taylor seres, ntroducng hgher-order terms 2 3 n n 1 x f '(x ) f ''(x ) f '''(x )... f (x ) Rn f (x ) f (x ) n 1 ( 1) R n f n ( ) 2! 3! n! n 1! For example, how can we derve the forward fnte-dfference estmate of the frst dervatve at x wth second order accuracy? f (x 1 ) f (x ) x f 2 3 f( x 1 ) f( x ) 2 '(x ) f ''(x ) O( ) f '(x ) f ''(x ) O( ) 2! 2! If we retan the second-dervatve, and estmate t wth frst-order accuracy, the order of accuracy for the estmate of f (x ) wll be p= Numercal Flud Mechancs PFJL Lecture 12, 19 19

20 FINITE DIFFERENCES: Taylor Seres, Hgher Order Accuracy Cont d 2! 3! n! 2 3 n Usng f (x ) f (x ) f '(x ) f ''(x ) f '''(x )... f n (x ) R 1 n n 1 ( 1) R n f n ( ) n 1! Estmate the second-dervatve wth forward fnte-dfferences at frstorder accuracy: 2 3 f( x 1 ) f( x ) x f '(x ) f ''(x ) O( ) 2! *( 2) f( x 2 ) 2 f( x 1 ) f( x ) 2 f ''(x ) O() *(1) f (x 2 ) f (x ) 2 f '(x ) f ''(x ) O( ) 2! f( x 1 ) f( x ) 2 f '( x ) f ''(x ) O( ) 2! f( x 1 ) f( x ) x f ( x 2 ) 2 f( x 1 ) f ( x ) 2 f ( x 2 ) 4 f( x 1 ) 3 f ( x ) 2 f '( x ) O ( x ) O ( ) 2 x 2! x 2 x 2.29 Numercal Flud Mechancs PFJL Lecture 12, 20 20

21 Fgure 23.1 Chapra and Canale Forward Dfferences 2.29 Numercal Flud Mechancs 21 PFJL Lecture 12, 21

22 Backward Dfferences 2.29 Numercal Flud Mechancs 22 PFJL Lecture 12, 22

23 Centered Dfferences 2.29 Numercal Flud Mechancs 23 PFJL Lecture 12, 23

24 FINITE DIFFERENCES Taylor Seres, Hgher Order Accuracy: EXAMPLE Problem: Estmate 1 st dervatve of f = -0.1*x^4-0.15*x^3-0.5*x^2-0.25*x +1.2 at x=0.5, wth a step sze of 0.25 and usng successvely hgher order schemes. How does the soluton mprove? %Defne the functon L11_FD.m f=@(x) -0.1*x^4-0.15*x^3-0.5*x^2-0.25*x +1.2; %Defne Step sze h=0.25; %Set pont at whch to evaluate the dervatve x = 0.5; %% Usng forward dfference %Frst order: df=(f(x+h)-f(x)) / h; fprntf('\n\n Frst order Forward dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) %Second order: df=(-f(x+2*h)+4*f(x+h)-3*f(x)) / (2*h); fprntf('second order Forward dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) %% Backwards dfference %Frst order: df=(-f(x-h)+f(x)) / (h); fprntf('frst order Backwards dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) %Second order: df=(f(x-2*h)-4*f(x-h)+3*f(x)) / (2*h); fprntf('second order Backwards dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) %% Central dfference %Second order: df=(f(x+h)-f(x-h)) / (2*h); fprntf('second order Central dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) %Fourth order: df=(-f(x+2*h)+8*f(x+h)-8*f(x-h)+f(x-2*h)) / (12*h); fprntf('fourth order Central dfference: %g, wth error:%g%% \n',df,abs(100*(df )/0.9125)) Output Frst order Forward dfference: , wth error: % Second order Forward dfference: , wth error: % Frst order Backwards dfference: , wth error: % Second order Backwards dfference: , wth error: % Second order Central dfference: , wth error: % Fourth order Central dfference: , wth error: e-014% Why s the 4 th order exact? 2.29 Numercal Flud Mechancs PFJL Lecture 12, 24 24

25 Approach: FINITE DIFFERENCES: Taylor Seres, Hgher Order Accuracy Summary Incorporate more hgher-order terms of the Taylor seres expanson than strctly needed and express them as fnte dfferences themselves e.g. for fnte dfference of m th dervatve at order of accuracy p, express the m+1 th, m+2 th, m+p-1 th dervatves at an order of accuracy p-1,.., 2, 1. General approxmaton: m s u au m j j r Can be used for forward, backward, skewed or central dfferences Can be computer automated Independent of coordnate system and extends to mult-dmensonal fnte dfferences (each coordnate s usually treated separately) Remember: order p of approxmaton ndcates how fast the error s reduced when the grd s refned (not the magntude of the error) 2.29 Numercal Flud Mechancs PFJL Lecture 12, 25 25

26 FINITE DIFFERENCES: Interpolaton Formulas for Hgher Order Accuracy 2 nd approach: Generalze Taylor seres usng nterpolaton formulas Ft the unknown functon soluton of the (P)DE to an nterpolaton curve and dfferentate the resultng curve. For example: Ft a parabola to data at ponts x 1, x, x 1 ( x x x 1 ), then dfferentate to obtan: f( x ) f( x ) f( x x f '( x ) 1 ( x 1 ) Ths s a 2 nd order approxmaton ) For unform spacng, reduces to centered dfference seen before In general, approxmaton of frst dervatve has truncaton error of the order of the polynomal All types of polynomals or numercal dfferentaton methods can be used to derve such nterpolatons formulas Polynomal fttng, Method of undetermned coeffcents, Newton s nterpolatng polynomals, Lagrangan and Hermte Polynomals, etc 2.29 Numercal Flud Mechancs PFJL Lecture 12, 26 26

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