2.29 Numerical Fluid Mechanics Spring 2015 Lecture 13

Transcription

1 REVIEW Lecture 12: Spring 2015 Lecture 13 Grid-Refinement and Error estimation Estimation of the order of convergence and of the discretization error Richardson s extrapolation and Iterative improvements using Roomberg s algorithm Fourier Error Analysis Provide additional information to truncation error: indicates how well Fourier mode solution, i.e. wavenumber and phase speed, is represented Effective wavenumber: e ikx i k e k k eff numerical k k k k eff ikx sin( k) nd (for CDS, 2 order, eff ) Effective wave speed (for linear convection eqn., f f c 0, integrating in time exactly): t num. dfk num. ikx ikeff t ik ( xceff t) ceff eff keff f ( t) c i k f ( x, t) f (0) e f (0) e dt c k (with i k c i k c ) eff eff eff PFJL Lecture 13, 1

2 REVIEW Lecture 12, Cont d: Stability Spring 2015 Lecture 13 Heuristic Method: trial and error n Energy Method: Find a quantity, l 2 norm 2, and then aim to show that it remains bounded for all n. Example: for c 0 t we obtained Von Neumann Method (Introduction), also called Fourier Analysis Method/Stability Hyperbolic PDEs and Stability ct nd order wave equation and waves on a string Characteristic finite-difference solution (review) ct Stability of C C (CDS in time/space, explicit): C 1 Example: Effective numerical wave numbers and dispersion PFJL Lecture 13, 2

3 FINITE DIFFERENCES Outline for Today Fourier Analysis and Error Analysis Differentiation, definition and smoothness of solution for order of spatial operators Stability Heuristic Method Energy Method Von Neumann Method (Introduction) : 1st order linear convection/wave eqn. Hyperbolic PDEs and Stability Example: 2nd order wave equation and waves on a string Effective numerical wave numbers and dispersion CFL condition: Definition Examples: 1st order linear convection/wave eqn., 2nd order wave eqn., other FD schemes Von Neumann examples: 1st order linear convection/wave eqn. Tables of schemes for 1st order linear convection/wave eqn. Elliptic PDEs FD schemes for 2D problems (Laplace, Poisson and Helmholtz eqns.) PFJL Lecture 13, 3

4 References and Reading Assignments Lapidus and Pinder, 1982: Numerical solutions of PDEs in Science and Engineering. Section 4.5 on Stability. Chapter 3 on Finite Difference Methods of J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3 rd edition, 2002 Chapter 3 on Finite Difference Approximations of H. Lomax, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, 2003 Chapter 29 and 30 on Finite Difference: Elliptic and Parabolic equations of Chapra and Canale, Numerical Methods for Engineers, 2014/2010/2006. PFJL Lecture 13, 4

5 Widely used procedure Von Neumann Stability Assumes initial error can be represented as a Fourier Series and considers growth or decay of these errors In theoretical sense, applies only to periodic BC problems and to linear problems Superposition of Fourier modes can then be used Again, use, ikx i x f ( x, t) fk ( t) e but for the error: ( x, t) ( t) e k Being interested in error growth/decay, consider only one mode: i x t i x ( t) e e e where is in general complex and function of : ( ) Strict Stability: The error will not to grow in time if t e 1 in other words, for t = nδt, the condition for strict stability can be written: e t t 1 or for e, 1 von Neumann condition Norm of amplification factor ξ smaller or equal to1 PFJL Lecture 13, 5

6 Evaluation of the Stability of a FD Scheme Von Neumann Example Consider again: c 0 t A possible FD formula ( upwind scheme) (t = nδt, x = Δx) which can be rewritten: n1 n n ct (1 ) 1 with x c t n1 n n n 1 Consider the Fourier error decomposition (one mode) and discretize it: (, ) ( ) i x t i x n x t t e e e e n t e i x Insert it in the FD scheme, assuming the error mode satisfies the FD (strictly valid for linear eq. only): (1 ) e e (1 ) e e e e n 1 n n ( n 1) t i x n t i x n t i ( 1) x 1 t 0-1 n+1 n x Cancel the common term (which is obtain: e (1 ) e t i e e n nt i ) in (linear) eq. and PFJL Lecture 13, 6

7 Evaluation of the Stability of a FD Scheme von Neumann Example The magnitude of complex conugate: Since t e is then obtained by multiplying ξ with its i i 2 i i e e (1 ) e (1 ) e 1 2 (1 ) 1 2 i i e e 2 cos( ) and 1 cos( ) 2sin ( ) 2 12 (1 ) 1 cos( ) 14 (1 )sin 2 ( ) 2 Thus, the strict von Neumann stability criterion gives (1 )sin ( ) 1 2 Since sin ( ) 0 1 cos( x) 0 2 we obtain the same result as for the energy method: c t c t 1 (1 ) ( ) Equivalent to the CFL condition PFJL Lecture 13, 7

8 Examples: Partial Differential Equations Hyperbolic PDE: B 2-4 A C > 0 (1) u 2 u c t (2) u u c 0 t (3) u ( U ) u g t (4) ( U) u g Wave equation, 2 nd order (from Lecture 10) Sommerfeld Wave/radiation equation, 1 st order Unsteady (linearized) inviscid convection (Wave equation first order) Steady (linearized) inviscid convection Allows non-smooth solutions Information travels along characteristics, e.g.: For (3) above: For (4), along streamlines: Domain of dependence of u(x,t) = characteristic path e.g., for (3), it is: x c (t) for 0< t < T Finite Differences, Finite Volumes and Finite Elements Upwind schemes d x dt c Ux ( ( t)) c d x c ds U t 0 x, y PFJL Lecture 13, 8

9 Waves on a String Partial Differential Equations Hyperbolic PDE - Example u( x, t) 2 u( x, t) c 0 x L, 0 t t Initial Conditions t (from Lecture 10) Boundary Conditions u0,t) ul,t) Wave Solutions u(x,0), u t (x,0) x Typically Initial Value Problems in Time, Boundary Value Problems in Space Time-Marching Solutions: Implicit schemes generally stable Explicit sometimes stable under certain conditions PFJL Lecture 13, 9

10 Wave Equation Partial Differential Equations Hyperbolic PDE - Example u( x, t) 2 u( x, t) c 0 x L, 0 t t t (from Lecture 10) Discretization: Finite Difference Representations (centered) u0,t) ul,t) +1-1 Finite Difference Representations i-1 i i+1 u(x,0), u t (x,0) x PFJL Lecture 13, 10

11 Partial Differential Equations Hyperbolic PDE - Example Introduce Dimensionless Wave Speed t (from Lecture 10) Explicit Finite Difference Scheme u0,t) ul,t) +1-1 Stability Requirement: ct C 1 Courant-Friedrichs-Lewy condition (CFL condition) i-1 i i+1 x u(x,0), u t (x,0) Physical wave speed must be smaller than the largest numerical wave speed, or, Time-step must be less than the time for the wave to travel to adacent grid points: PFJL Lecture 13, 11 c t or t c

12 Partial Differential Equations Hyperbolic PDE - Example Start of Integration: Euler and Higher Order Starters Given ICs: t 1 st order Euler Starter But, second derivative in x at t = 0 is known from IC: From Wave Equation u0,t) ul,t) Higher order Taylor Expansion +k Higher Order Self Starter =2-2 i-1 i i+1 u(x,0)=f(x), u t (x,0)=g(x) =1 x General idea: use the PDE itself to get higher order integration PFJL Lecture 13, 12

13 Waves on a String u( x, t) 2 u( x, t) c 0 x L, 0 t t Initial condition L=10; T=10; waveeq.m c=1.5; N=100; h=l/n; M=400; k=t/m; C=c*k/h Lf=0.5; x=[0:h:l]'; t=[0:k:t]; %fx=['exp(-0.5*(' num2str(l/2) '-x).^2/(' num2str(lf) ').^2)']; %gx='0'; fx='exp(-0.5*(5-x).^2/0.5^2).*cos((x-5)*pi)'; gx='0'; %Zero first time derivative at t=0 f=inline(fx,'x'); g=inline(gx,'x'); n=length(x); m=length(t); u=zeros(n,m); % Second order starter u(2:n-1,1)=f(x(2:n-1)); for i=2:n-1 u(i,2) = (1-C^2)*u(i,1) + k*g(x(i)) +C^2*(u(i-1,1)+u(i+1,1))/2; end % CDS: Iteration in time () and space (i) for =2:m-1 for i=2:n-1 u(i,+1)=(2-2*c^2)*u(i,) + C^2*(u(i+1,)+u(i-1,)) - u(i,-1); end end figure(1) plot(x,f(x)); a=title(['fx = ' fx]); set(a,'fontsize',16); figure(2) wavei(u',x,t); a=xlabel('x'); set(a,'fontsize',14); a=ylabel('t'); set(a,'fontsize',14); a=title('waves on String'); set(a,'fontsize',16); colormap; PFJL Lecture 13, 13

14 Waves on a String, Longer simulation: Effects of dispersion and effective wavenumber/speed L=10; T=10; waveeq.m c=1.5; N=100; h=l/n; % Horizontal resolution (Dx) M=400; % Test: increase duration of simulation, to see effect of %dispersion and effective wavenumber/speed (due to 2 nd order) %T=100;M=4000; k=t/m; % Time resolution (Dt) C=c*k/h % Try case C>1, e.g. decrease Dx or increase Dt Lf=0.5; x=[0:h:l]'; t=[0:k:t]; %fx=['exp(-0.5*(' num2str(l/2) '-x).^2/(' num2str(lf) ').^2)']; %gx='0'; fx='exp(-0.5*(5-x).^2/0.5^2).*cos((x-5)*pi)'; gx='0'; f=inline(fx,'x'); g=inline(gx,'x'); n=length(x); m=length(t); u=zeros(n,m); %Second order starter u(2:n-1,1)=f(x(2:n-1)); for i=2:n-1 u(i,2) = (1-C^2)*u(i,1) + k*g(x(i)) +C^2*(u(i-1,1)+u(i+1,1))/2; end %CDS: Iteration in time () and space (i) for =2:m-1 for i=2:n-1 u(i,+1)=(2-2*c^2)*u(i,) + C^2*(u(i+1,)+u(i-1,)) - u(i,-1); end end PFJL Lecture 13, 14

15 Wave Equation d Alembert s Solution Wave Equation u( x, t) 2 u( x, t) c t 0 x L, 0 t Solution t Periodicity Properties F u0,t) G F ul,t) Proof u(x,0)=f(x), u t (x,0)=g(x) x PFJL Lecture 13, 15

16 Hyperbolic PDE Method of Characteristics Explicit Finite Difference Scheme t First 2 Rows known u0,t) G F ul,t) +1-1 Characteristic Sampling Exact Discrete Solution u(x,0)=f(x), u t (x,0)=g(x) x PFJL Lecture 13, 16

17 Hyperbolic PDE Method of Characteristics Let s proof the following FD scheme is an exact Discrete Solution D Alembert s Solution with C=1 t u0,t) ul,t) G F +1-1 Proof u(x,0)=f(x), u t (x,0)=g(x) x PFJL Lecture 13, 17

18 Courant-Fredrichs-Lewy Condition (1920 s) Basic idea: the solution of the Finite-Difference (FD) equation can not be independent of the (past) information that determines the solution of the corresponding PDE In other words: The Numerical domain of dependence of FD scheme must include the mathematical domain of dependence of the corresponding PDE time CFL NOT satisfied time CFL satisfied uxt (,) uxt (,) PDE dependence Numerical stencil space space PFJL Lecture 13, 18

19 Determine domain of dependence of PDE and of FD scheme PDE: FD scheme. For our Upwind discretization, with t = nδt, x = Δx : => CFL condition: CFL: Linear convection (Sommerfeld Eqn) Example u( x, t) u( x, t) c t 0 Solution of the form: u( x, t) F( x ct) dt 1 Slope of characteristic: dx c t Slope of Upwind scheme: ct 1 t 1 c dx Characteristics: If c x c t and du 0 u cst dt c t n1 n n n 1 True solution is outside of numerical domain of influence 0 CFL NOT satisfied -1 CFL satisfied n+1 n True solution is within numerical domain of influence Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Source: Figure 2.1 from Durran, D. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, PFJL Lecture 13, 19

20 CFL: 2 nd order Wave equation Example Determine domain of dependence of PDE and of FD scheme PDE, second order wave eqn example: u( x, t) 2 u( x, t) c 0 x L, 0 t t As seen before: u( x, t) F( x ct) G( x ct) slope of characteristics: FD scheme: discretize: t = nδt, x = Δx dt dx 1 c CD scheme (CDS) in time and space (2nd order), explicit n1 n n1 n n n u 2u u 2 u 1 2u u 1 n1 2 n 2 n n n1 ct c u ( ) ( C u C u 1 u 1) u where C= t We obtain from the respective slopes: t ct 1 Full line case: CFL satisfied Dotted lines case: c and Δt too big, Δx too small (CFL NOT satisfied) n x PFJL Lecture 13, 20

21 CFL Condition: Some comments CFL is only a necessary condition for stability Other (sufficient) stability conditions are often more restrictive u( x, t) u( x, t) t For example: if c 0 is discretized as u( x, t) u( x, t) c t nd One obtains from the CFL: CD,2 order in t CD,4 order in x ct 2 While a Von Neumann analysis leads: th ct Five grid-points stencil: (-1,8,0,-8,1) / 12 See Taylor tables in previous lecture and eqn. sheet For equations that are not purely hyperbolic or that can change of type (e.g. as diffusion term increases), CFL condition can at times be violated locally for a short time, without leading to global instability further in time PFJL Lecture 13, 21

22 von Neumann Examples Forward in time (Euler), centered in space, Explicit n1 n n n 1 1 n1 n C n n c 0 ( 1 1) t 2 2 Von Neumann: insert n1 n C n n t C i i ( 1 1) e 1 e e 1 Ci sin( ) Taking the norm: Unconditionally Unstable Implicit scheme (backward in time, centered in space) Unconditionally Stable ( x, t) ( t) e e e e e i x t i x n n t i t e 1 Ci sin( ) 1 Ci sin( ) 1 C sin ( ) 1 for C 0! n1 n n1 n1 1 1 n1 n C n1 n1 t t c 0 ( 1 1) e 1 e Ci sin( ) t 2 2 t 1 1 e 1 for C 0! 1 Ci sin( ) 1 Ci sin( ) 1 C sin ( ) PFJL Lecture 13, 22

23 Stability of FD schemes for u t + b u y = 0 (t denoted x below) Table showing various finite difference forms removed due to copyright restrictions. Please see Table 6.1 in Lapidus, L., and G. Pinder. Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley-Interscience, PFJL Lecture 13, 23

24 Stability of FD schemes for u t + b u y = 0, Cont. Table showing various finite difference forms removed due to copyright restrictions. Please see Table 6.1 in Lapidus, L., and G. Pinder. Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley-Interscience, PFJL Lecture 13, 24

25 Partial Differential Equations Elliptic PDE (from Lecture 9) Laplace Operator Examples: ϕ ϕ Laplace Equation Potential Flow Poisson Equation Potential Flow with sources Steady heat conduction in plate + source 2 Uu u Helmholtz equation Vibration of plates Steady Convection-Diffusion Smooth solutions ( diffusion effect ) Very often, steady state problems Domain of dependence of u is the full domain D(x,y) => global solutions Finite differ./volumes/elements, boundary integral methods (Panel methods) PFJL Lecture 13, 25

26 Partial Differential Equations Elliptic PDE - Example (from Lecture 9) y Equidistant Sampling u(x,b) = f 2 (x) Discretization Finite Differences u0,y)=g 1 (y) ua,y)=g 2 (y) +1-1 i-1 i i+1 u(x,0) = f 1 (x) x Dirichlet BC PFJL Lecture 13, 26

27 Partial Differential Equations Elliptic PDE - Example (from Lecture 9) Discretized Laplace Equation y Finite Difference Scheme u(x,b) = f 2 (x) Boundary Conditions i u0,y)=g 1 (y) ua,y)=g 2 (y) +1-1 i Global Solution Required i-1 i i+1 u(x,0) = f 1 (x) x PFJL Lecture 13, 27

28 Elliptic PDEs Laplace Equation, Global Solvers p 7 p 8 p 9 p 4 p 5 p 6 Dirichlet BC Leads to Ax = b, with A block-tridiagonal: A = tri { I, T, I } u 1,2 p 1 p 2 p 3 u 5,2 u 2,1 u 5,2 PFJL Lecture 13, 28

29 Ellipticic PDEs Neumann Boundary Conditions Neumann (Derivative) Boundary Condition given Finite Difference Scheme at i = n y Finite Difference Scheme u(x,b) = f 2 (x) Derivative BC: Finite Difference u0,y)=g 1 (y) u x a,y) given Boundary Finite Difference Scheme at i = n u un 1, 2 un 1, un, 1 un, 1 4un, 0 n i-1 i i+1 u(x,0) = f 1 (x) Leads to a factor 2 (a matrix 2 I in A) for points along boundary x PFJL Lecture 13, 29

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