Finite Dierence Schemes

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1 MATH-459 Numerical Methods for Coservatio Laws by Prof. Ja S. Hesthave Solutio set 2: Fiite Dierece Schemes Exercise 2. Cosistecy A method is cosistet if its local trucatio error T k satises T k (x, t) = O (k p ) + O (h q ) where p, q >. () Essetially, cosistecy tells us that we are approximatig the solutio of the correct PDE. Covergece A method is coverget if the error E k satises E k (, t) as k, t. (2) Stability A method + = H k is stable if for each T > there exist costats C ad k, such that H k C k T, < k < k. (3) The importace of stability is made clear by the Lax equivalece theorem. I class we saw a proof of oe directio of this equivalece: if a method + = H k is cosistet ad stable, the it is coverget. From the proof it is clear that stability allows us, i priciple, to approximate a solutio at ay ite time iterval [, T ], ad to ay degree of accuracy. Exercise 2.2 (b) is give by Let u be a smooth solutio of u t + au x =. The local trucatio error for the Leapfrog scheme kt k (x, t) = u(x, t + k) u(x, t k) + ak h (u (x + h, t) u (x h, t)). (4) Next, we expad all the terms o the right had side of the last equatio about (x, t). For example, we have u(x, t + k) ad u(x, t k) give by u(x, t + k) = u + u t k + 2 u ttk 2 + O ( k 3) (5) ad u(x, t k) = u u t k + 2 u ttk 2 + O ( k 3), (6) where for simplicity of otatio we have replaced u (x, t), u t (x, t) ad u tt (x, t) by u, u t ad u tt, respectively. It is, thus, clear that u(x, t + k) u(x, t k) = 2ku t + O ( k 3). (7) By repeatig the calculatio also for the shifts i space u(x ± h, t), we get kt k (x, t) = 2ku t + O ( k 3) + ak h ( 2hux + O ( h 3)), (8) which implies T k (x, t) = 2 (u t + au x ) + O ( k 2) + O ( h 2). (9) Sice u satises u t + au x =, the local trucatio error is simply O ( k 2) + O ( h 2). (c) Observe that at the th time level, to calculate +, the Leapfrog scheme requires the values, ot oly of, but also of. Compared to the LF ad LW schemes which use oly to calculate +, this is a clear disadvatage. I computatios, keepig the umerical solutio at more tha oe time level multiplies the size of memory required for the program.

2 Exercise 2.3 (a) To show that the LF scheme is stable provided a k h, () we will show that H is bouded for all, where H is the operator satisfyig + = H. To do so, we calculate + = h + j = h ( 2 j+ + ak ( j ) 2h j+ ) j j j h ( ak 2 h j+ + + ak h ) j j ( ak 2 h + + ak ) h. () Owig to the calculatio above, wheever () holds, we have H = + (2) ad therefore H H =, 2,.... (3) (b), (c) The CFL coditio requires that the domai of depece of the coservatio law is cotaied i the umerical domai of depece. Therefore, the CFL coditio is ecessary for stability, however, as the example i (e) shows, it is ot suciet. (d) Sice the CFL coditio is ecessary for stability, ad () esures stability, it is clear that () implies that the CFL coditio is satised. Notice that i this example, the CFL coditio is i fact ecessary ad suciet for stability. (e) The CFL coditio is ot always suciet for stability. For example The CFL coditio of the scheme + j = j ak ( 2h j+ j ) is give by a k/h, the same as the CFL of the LF scheme. However, we kow that (4) is ucoditioally ustable. Exercise 2.4 (a) Matlab code for implemetig the schemes to solve the advectio ca be foud o the last two pages of this solutio maual. (b) I Figure, u (x, t =.5)) is plotted as solved by the pwid, Lax- Friedrichs, Lax-Wro ad Beam-Warmig methods. (c) Both pwid ad Lax-Friedrichs catch the jump, but we observe less umerical dissipatio from the upwid scheme. This is because the upwid scheme exploits that iformatio is oly movig i oe directio. The higher order methods Lax-Wro ad Beam-warmg both itroduce oscillatios aroud the discotiuities. (d,e) See Figure 2. (f ) Notice how, o this problem with o-smooth solutios, the rate of covergece of the rst order methods is ow O ( h /2), while the rate of covergece of the secod order methods are O ( h 2/3) at best. Such a low order of accuracy is ot practical for the solutio of real problems, ad i the ext lecture other methods that hadle discotiuities better are itroduced. (4) 2

3 .5 Lax-Friedrichs.5 pwid (a) (b).5 Lax-Wro.5 Beam-Warmig (c) (d) Figure : The aalytical solutio ad the umerical solutio u(x, t =.5) for the advectio equatio o Riema iitial data with a =, h =.5, k h =.5. Lax-Friedrichs. Slope = pwid. Slope = (a) (b) Lax-Wro. Slope = Beam-Warmig. Slope = (c) (d) Figure 2: The error as a fuctio of resolutio. The error is measured i -orm, the resolutio is measured as the umber of discretizatio poits i the x dimesio. 3

4 % solutio2.m clear all,clc % This script was writte for EPFL MATH459, Numerical Methods for Coservatio Laws % ad tested with Octave The scalar advectio equatio du/dx + a * du/dt = % with riema iitial data is solved usig a Leapfrog, pwid, Lax Friedirch, Lax % Wroff or Beam Warmig umerical scheme. % The solutio is visualized ad the accuracy of the scheme tested. % Choose SolverNumber % : pwid % 2: LaxFriedrich % 3: LaxWroff % 4: BeamWarmig SolverNumber = ; PlotSolutio = ; TestAccuracy = ; % Fuctio declaratio fuctio [A] = pwid(c,x) A = diag(c*oes(x,), )+diag(( C)*oes(X,),); fuctio [A] = LaxFriedrich(C,X) A = diag(.5*(+c)*oes(x,), )+diag(.5*( C)*oes(X,),); fuctio [A] = LaxWroff(C,X) A = diag(.5*(c^2+c)*oes(x,), )+diag(( C^2)*oes(X,),)+... diag(.5*(c^2 C)*oes(X,),); fuctio [A] = BeamWarmig(C,X) A = diag((.5*c+.5*c^2)*oes(x,),)+diag((2*c C^2)*oes(X,), )+... diag((.5*c+.5*c^2)*oes(x 2,), 2); fuctio [A,ame] = GetMatrix(SolverNumber,C,X) switch SolverNumber case A = pwid(c,x); ame = 'pwid'; case 2 A = LaxFriedrich(C,X); ame = 'Lax Friedrichs'; case 3 A = LaxWroff(C,X); ame = 'Lax Wroff'; case 4 A = BeamWarmig(C,X); ame = 'Beam Warmig'; 4

5 % Visualizatio if PlotSolutio a = ; % Discretizatio h =.25; k =.5*h; X = :h:; X = umel(x); T = :k:.5; T = umel(t); % Geerate correct solutio u = zeros(x,t); for i = :T u(fid(x<=a*t(i)),i) = ; % Solve usig the umerical scheme [A,ame] = GetMatrix(SolverNumber,a*k/h,X); = zeros(x,t); (:,) = u(:,); for i = :T (:,i+) = A*(:,i); (:2,i+) = ; % Make a plot for i = :T plot(x,(:,i),' r',x,u(:,i),' b'); ylim([.5.5]);grid o;title(ame); drawow % Accuracy test if TestAccuracy H = [.8,.4,.2,.,.5,.25,.25]; E = zeros(umel(h),); = zeros(umel(h),); % Fid Error for i = :umel(h) a = ; % Discretizatio h = H(i); k =.5*h; X = :h:; X = umel(x); T = :k:.5; T = umel(t); % Geerate correct solutio u = zeros(x,); u(fid(x<=a*t())) = ; % Solve usig the umerical scheme = zeros(x,);(fid(x<=)) = ; [A,ame] = GetMatrix(SolverNumber,a*k/h,X); for j = 2:T = A*; (:2) = ; % Measure error i orm E(i) = sum(abs((:,) u))/x; (i) = X; % Make loglog plot p = polyfit(log(),log(e),); loglog(,e,' ok');grid o; xlabel('resolutio');ylabel('error'); title([ame,'. Slope = ',um2str(p())]); 5

Computational Fluid Dynamics. Lecture 3

Computational Fluid Dynamics. Lecture 3 Computatioal Fluid Dyamics Lecture 3 Discretizatio Cotiued. A fourth order approximatio to f x ca be foud usig Taylor Series. ( + ) + ( + ) + + ( ) + ( ) = a f x x b f x x c f x d f x x e f x x f x 0 0