Characteristic finite-difference solution Stability of C C (CDS in time/space, explicit): Example: Effective numerical wave numbers and dispersion

Transcription

1 Spring 015 Lecture 14 REVIEW Lecture 13: Stability: Von Neumann Ex.: 1st order linear convection/wave eqn., F-B scheme Hyperbolic PDEs and Stability nd order wave equation and waves on a string Characteristic finite-difference solution Stability of C C (CDS in time/space, explicit): Example: Effective numerical wave numbers and dispersion CFL condition: Numerical domain of dependence must include Mathematical domain of dependence Examples: 1st order linear convection/wave eqn., nd order wave eqn. Other FD schemes (C nd C 4 th ) Von Neumann: 1 st order linear convection/wave eqn., F- C: unstable Stability summary: Tables of schemes for 1 st order linear convection/wave eqn. Elliptic PDEs FD schemes for D problems (Laplace, Poisson and Helmholtz eqns.) Direct nd order and Iterative (Jacobi, Gauss-Seidel) Boundary conditions ct C 1 x ct C 1 x PFJL Lecture 14, 1

2 Elliptic PDEs, Continued TODAY (Lecture 14): FINITE DIFFERENCES, Cont d Examples, Higher order finite differences Irregular boundaries: Dirichlet and Von Neumann BCs Internal boundaries Parabolic PDEs and Stability Explicit schemes (1D-space) Von Neumann Implicit schemes (1D-space): simple and Crank-Nicholson Von Neumann Examples Extensions to D and 3D Explicit and Implicit schemes Alternating-Direction Implicit (ADI) schemes PFJL Lecture 14,

3 TODAY (Lecture 14, Cont d): FINITE VOLUME METHODS Integral forms of the conservation laws Introduction to FV Methods Approximations needed and basic elements of a FV scheme FV grids Approximation of surface integrals (leading to symbolic formulas) Approximation of volume integrals (leading to symbolic formulas) Summary: Steps to step-up FV scheme Examples: One Dimensional examples Generic equations Linear Convection (Sommerfeld eqn.): convective fluxes nd order in space, 4 th order in space, links to CDS Unsteady Diffusion equation: diffusive fluxes Two approaches for nd order in space, links to CDS PFJL Lecture 14, 3

4 References and Reading Assignments Lapidus and Pinder, 198: 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, 00 Chapter 3 on Finite Difference Approximations of H. Lomax, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, 003 Chapter 9 and 30 on Finite Difference: Elliptic and Parabolic equations of Chapra and Canale, Numerical Methods for Engineers, 014/010/006. PFJL Lecture 14, 4

5 Elliptic PDEs Iterative Schemes: Laplace equation Finite Difference Scheme u u u u u k k k k k 1 i1, i1, i, 1 i, 1 4 i, 0 Liebman Iterative Scheme (Jacobi/Gauss-Seidel) k1 u u u u ui, 4 u x k k k k i1, i1, i, 1 i, 1 u y 0 r k i, u u u u 4u 4 k k k k k i1, i1, i, 1 i, 1 i, SOR Iterative Scheme, Jacobi y u(x,b) = f (x) k u u u u 4u ui, 4 k k k k k u u u u (1 ) ui, 4 Optimal SOR (Equidistant Sampling h) k k k k k i 1, i 1, i, 1 i, 1 i, i1, i1, i, 1 i, 1 u(0,y)=g 1 (y) u(a,y)=g (y) +1-1 i-1 i i+1 u(x,0) = f 1 (x) x PFJL Lecture 14, 5

6 Elliptic PDE: Poisson Equation SOR Iterative Scheme, with Jacobi k u u u u 4u h g ui, 4 k u u u u h g (1 ) ui, 4 k k k k k i1, i1, i, 1 i, 1 i, i, k k k k i1, i1, i, 1 i, 1 i, PFJL Lecture 14, 6

7 Elliptic PDE: Poisson Equation SOR Iterative Scheme, with Gauss-Seidel k u u u u 4u h g ui, 4 k u u u u h g (1 ) ui, 4 k k+1 k k+1 k i1, i1, i, 1 i, 1 i, i, k k+1 k k+1 i1, i1, i, 1 i, 1 i, PFJL Lecture 14, 7

8 Laplace Equation Steady Heat diffusion (with source: Poisson eqn) Lx=1; Ly=1; N=10; h=lx/n; M=floor(Ly/Lx*N); niter=0; eps=1e-6; x=[0:h:lx]'; y=[0:h:ly]; f1x='4*x-4*x.^'; %f1x='0' fx='0'; g1x='0'; gx='0'; gxy='0'; f1=inline(f1x,'x'); f=inline(fx,'x'); g1=inline(g1x,'y'); g=inline(gx,'y'); gf=inline(gxy,'x','y'); n=length(x); m=length(y); u=zeros(n,m); u(:n-1,1)=f1(x(:n-1)); u(:n-1,m)=f(x(:n-1)); u(1,1:m)=g1(y); u(n,1:m)=g(y); for i=1:n for =1:m g(i,) = gf(x(i),y()); end end duct.m u_0=mean(u(1,:))+mean(u(n,:))+mean(u(:,1))+mean(u(:,m)); u(:n-1,:m-1)=u_0*ones(n-,m-); omega=4/(+sqrt(4-(cos(pi/(n-1))+cos(pi/(m-1)))^)) for k=1:niter u_old=u; for i=:n-1 for =:m-1 u(i,)=(1-omega)*u(i,) +omega*(u(i-1,)+u(i+1,)+u(i,-1)+u(i,+1)-h^*g(i,))/4; end end r=abs(u-u_old)/max(max(abs(u))); k,r if (max(max(r))<eps) break; end end figure(3) surf(y,x,u); shading interp; a=ylabel('x'); set(a,'fontsize',14); a=xlabel('y'); set(a,'fontsize',14); a=title(['poisson Equation - g = ' gxy]); set(a,'fontsize',16); u u gxy (, ) x y BCs : u ( x,0, t) f ( x) 4x 4x Three other BCs are null PFJL Lecture 14, 8

9 Helmholtz Equation SOR Iterative Scheme, with Gauss-Seidel u k k +1 k k +1 k k ui 1, ui 1, ui, 1 ui, 1 (4 h fi, ) ui, h gi, i, (4 h fi, ) (1 ) u k k +1 k k+1 k ui 1, ui 1, ui, 1 ui, 1 h gi, i, (4 h fi, ) PFJL Lecture 14, 9

10 Elliptic PDE s Higher Order Finite Differences CD, 4 th order (see tables in eqs. sheet) y u 16 u 30 u 16 u u k k k k k u i, i1, i, i1, i, x th 1h CD, 4 order u(x,b) = f (x) The resulting 9 point cross stencil (not drawn) is more challenging computationally (boundary, etc) than CD nd order. u(0,y)=g 1 (y) Use more compact scheme instead u(a,y)=g (y) +1-1 Square stencil (see figure): Use Taylor series, then cancel the terms so as to get a 4 th order scheme i-1 i i+1 u(x,0) = f 1 (x) x Leads to: + PFJL Lecture 14, 10

11 Elliptic PDEs: Irregular Boundaries Many elliptic problems don t have simple boundaries/geometries One way to handle them is through irregular discrete boundary cells (e.g. shaved cells) 1) Boundary Stencils (with Dirichlet BCs) i-1, α 1 x β y α x 1 st derivatives evaluated at center of edges, hence dx is sum of half edge lengths on each side i-1, i y, x, i i, -1 β 1 y Image of a grid for a heated plate that has an irregularly shaped boundary. Image by MIT OpenCourseWare. Can be used directly with Dirichlet BCs Leads to direct and iterative elliptic solvers as before, but with updated coefficients for the boundary stencils Other options possible: curved boundary elements PFJL Lecture 14, 11

12 Elliptic PDEs: Irregular Boundaries ) Neumann Boundary Conditions (e.g. normal derivative given) 3 θ y 7 Linear interpolation at 7 6 x Curved boundary in which the normal gradient is specified. Image by MIT OpenCourseWare. 5 This is an approach given in Chapra & Canale One may instead estimate u 3 from neighbor nodes, then take the derivative along 1-3 PFJL Lecture 14, 1

13 Elliptic PDEs Internal (Fixed) Boundaries Velocity and Stress Continuity (heat flux or viscous stress) Derivative Finite Differences (1 st order) McGraw-Hill. All rights reserved. This content is excluded fromourcreativecommons license. For more information, seehttp://ocw.mit.edu/fairuse. Source: Chapra, S. and R. Canale. Numerical Methods for Engineers.McGraw-Hill, 005. y Finite Difference Equation at bnd. i, SOR Finite Difference Scheme at bnd i, i, i-1 i i+1 x PFJL Lecture 14, 13

14 Elliptic PDEs Internal (Fixed) Boundaries Higher Order Velocity and Stress Continuity Taylor Series, inserting the PDE u u f ( x, y) f ( x, y), g ( x, y) x y y g i, xx g i, Derivative Finite Differences ( nd order) xx +1-1 g i, Finite Difference Equation at bnd. SOR Finite Difference Scheme at bnd. g i, g i, g i, g g i-1 i i+1 McGraw-Hill. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Source: Chapra, S. and R. Canale. Numerical Methods for Engineers. McGraw-Hill, 005. g i, i, i, fi, x PFJL Lecture 14, 14

15 Partial Differential Equations Parabolic PDE: B - 4 A C = 0 (from Lecture 9) Examples T T f, ( ) t c c u u t g Heat conduction equation, forced or not (dominant in 1D) Unsteady, diffusive, small amplitude flows or perturbations (e.g. Stokes Flow) Usually smooth solutions ( diffusion effect present) t Propagation problems Domain of dependence of solution is domain D ( x, y, and 0 < t < ): BC 1: T(0,0,t)=f 1 (t) D( x, y, 0 < t < ) BC : T(L x,l y,t)=f (t) Finite Differences/Volumes, Finite Elements IC: T(x,y,0)=F(x,y) PFJL Lecture 14, 15 0 L x,l y x, y

16 (from Lecture 9) Partial Differential Equations Parabolic PDE: 1D Heat Conduction example Heat Conduction Equation ( ( T x, t T x, t,0 x L,0 t t xx Insulation Initial Condition ( Boundary Conditions s c T x,0 f ( x),0 x L T(0, t) g ( t),0 t 1 T( L, t) g ( t),0 t x = 0 T(0,t) = g 1 (t) Rod x = L T(L,t)=g (t) x PFJL Lecture 14, 16

17 Parabolic PDE 1D Heat Conduction: Forward in time, centered in space, explicit (from Lecture 9) Discretization Equidistant Sampling t F-C: computational stencil Forward (Euler) Finite Difference in time T( xi, t 1) T( xi, t ) T(0,t)=g 1 (t) Tt ( x, t) O( k) k Centered Finite Difference in space T( xi 1, t ) T( xi, t ) T( xi 1, t ) Txx( x, t) O( h ) h Ti, T( xi, t ) Finite Difference Equation T T T T T k h i, 1 i, i 1, i, i 1, i-1 i i+1 T(x,0) = f(x) T(L,t)=g (t) +1-1 x PFJL Lecture 14, 17

18 Parabolic PDE 1D Heat Conduction: Forward in time, centered in space, explicit Dimensionless diffusion coefficient Explicit Finite Difference Scheme ( k r h T 1 r T r ( T T ) i, 1 i, i1, i1, t F-C: computational stencil Stability Requirement r <= 0.5 T(0,t)=g 1 (t) T(L,t)=g (t) +1-1 Conditionally stable (von Neumann) Shown in class on blackboard i-1 i i+1 T(x,0) = f(x) x PFJL Lecture 14, 18

19 Heat Conduction Equation Explicit Finite Differences (1D-in-space, unsteady case; similar to steady elliptic problem seen previously) T denoted by u, i.e. α denoted by c, i.e. u c i, i, T L=1; T=0.; c=1; N=5; h=l/n; M=10; k=t/m; r=c^*k/h^ heat_fw.m x=[0:h:l]'; t=[0:k:t]; fx='4*x-4*x.^'; g1x='0'; gx='0'; f=inline(fx,'x'); g1=inline(g1x,'t'); g=inline(gx,'t'); n=length(x); m=length(t); u=zeros(n,m); u(:n-1,1)=f(x(:n-1)); u(1,1:m)=g1(t); u(n,1:m)=g(t); for =1:m-1 for i=:n-1 u(i,+1)=(1-*r)*u(i,) + r*(u(i+1,)+u(i-1,)); end end. figure(4) mesh(t,x,u); a=ylabel('x'); set(a,'fontsize',14); a=xlabel('t'); set(a,'fontsize',14); a=title(['forward Euler - r =' numstr(r)]); set(a,'fontsize',16); ICs: BCs: 0. PFJL Lecture 14, 19

20 Heat Conduction Equation Explicit Finite Differences L=1; T=0.333; c=1; N=5; h=l/n; M=10; k=t/m; r=c^*k/h^ heat_fw_.m ICs: 0.33 x=[0:h:l]'; t=[0:k:t]; fx='4*x-4*x.^'; g1x='0'; gx='0'; f=inline(fx,'x'); g1=inline(g1x,'t'); g=inline(gx,'t'); n=length(x); m=length(t); u=zeros(n,m); u(:n-1,1)=f(x(:n-1)); u(1,1:m)=g1(t); u(n,1:m)=g(t); for =1:m-1 for i=:n-1 u(i,+1)=(1-*r)*u(i,) + r*(u(i+1,)+u(i-1,)); end end. figure(4) mesh(t,x,u); a=ylabel('x'); set(a,'fontsize',14); a=xlabel('t'); set(a,'fontsize',14); a=title(['forward Euler - r =' numstr(r)]); set(a,'fontsize',16); BCs: PFJL Lecture 14, 0

21 Parabolic PDE: Implicit Schemes Leads to a system of equations to be solved at each time-step B-C (Backward-Centered): 1 st order accurate in time, nd order in space Unconditionally stable x i-1 Simple implicit method t l+1 t l x i x i+1 Grid point involved in time difference Grid point involved in space difference B-C: Backward in time Centered in space Evaluates RHS at time t+1 instead of time t (for the explicit scheme) Crank-Nicolson: nd order accurate in time, nd order in space Unconditionally stable x i-1 Crank-Nicolson method t l+1 t l+1/ t l x i x i+1 Grid point involved in time difference Grid point involved in space difference Time: centered FD, but evaluated at mid-point nd derivative in space determined at mid-point by averaging at t and t+1 Image by MIT OpenCourseWare. After Chapra, S., and R. Canale. Numerical Methods for Engineers. McGraw-Hill, 005. PFJL Lecture 14, 1

22 Parabolic PDE: Implicit Schemes Crank-Nicolson Scheme Equidistant Sampling t Discretization Mid-point Finite Difference in time u(0,t)=g 1 (t) u(l,t)=g (t) +1 Finite Difference Equation Crank-Nicholson Implicit Scheme i-1 i i+1 u(x,0) = f(x) x Unconditionally stable (by Von Neumann) PFJL Lecture 14,

23 Parabolic PDEs: Implicit Schemes Crank-Nicolson special case of r = 1 +1 PFJL Lecture 14, 3

24 L=1; T=0.333; c=1; N=5; h=l/n; M=10; k=t/m; r=c^*k/h^ Heat Flow Equation Implicit Crank-Nicolson Scheme heat_cn.m ICs: x=[0:h:l]'; t=[0:k:t]; fx='4*x-4*x.^'; g1x='0'; gx='0'; f=inline(fx,'x'); g1=inline(g1x,'t'); g=inline(gx,'t'); n=length(x); m=length(t); u=zeros(n,m); u(:n-1,1)=f(x(:n-1)); u(1,1:m)=g1(t); u(n,1:m)=g(t); % set up Crank-Nicholson coef matrix d=(+*r)*ones(n-,1); b=-r*ones(n-,1); c=b; % LU factorization [alf,bet]=lu_tri(d,b,c); for =1:m-1 rhs=r*(u(1:n-,)+u(3:n,)) +(-*r)*u(:n-1,); rhs(1) = rhs(1)+r*u(1,+1); rhs(n-)=rhs(n-)+r*u(n,+1); % Forward substitution z=forw_tri(rhs,bet); % Back substitution y_b=back_tri(z,alf,c); for i=:n-1 u(i,+1)=y_b(i-1); end end BCs: PFJL Lecture 14, 4

25 L=1; T=0.1; c=1; N=10; h=l/n; M=10; k=t/m; r=c^*k/h^ Heat Flow Equation Implicit Crank-Nicolson Scheme heat_cn_sin.m x=[0:h:l]'; t=[0:k:t]; fx='sin(pi*x)+sin(3*pi*x)'; g1x='0'; gx='0'; f=inline(fx,'x'); g1=inline(g1x,'t'); g=inline(gx,'t'); n=length(x); m=length(t); u=zeros(n,m); u(:n-1,1)=f(x(:n-1)); u(1,1:m)=g1(t); u(n,1:m)=g(t); % set up Crank-Nicholson coef matrix d=(+*r)*ones(n-,1); b=-r*ones(n-,1); c=b; % LU factorization [alf,bet]=lu_tri(d,b,c); for =1:m-1 rhs=r*(u(1:n-,)+u(3:n,)) +(-*r)*u(:n-1,); rhs(1) = rhs(1)+r*u(1,+1); rhs(n-)=rhs(n-)+r*u(n,+1); % Forward substitution z=forw_tri(rhs,bet); % Back substitution y_b=back_tri(z,alf,c); for i=:n-1 u(i,+1)=y_b(i-1); end end Initial Condition Analytical Solution PFJL Lecture 14, 5

26 Parabolic PDEs: Two spatial dimensions Example: Heat conduction equation/unsteady diffusive (e.g. negligible flow, no convection) Standard explicit and implicit schemes (t = nδt, x = iδx, y= Δy) n1 n n n n n n n Ti, Ti, Ti 1, Ti, Ti 1, Ti, 1 Ti, Ti, 1 c c O( t), O( x y ) t x y Stringent stability criterion: 1 x y tc 1 t For uniform grid: r 8 c x 4 Explicit: Implicit: T T T c, (0 t, 0 x L, 0 ) x y Ly t x y T T T T T T T T Crank-Nicolson Implicit (for Δx=Δy) O( t ), O( x ) n1 n r n1 n1 n1 n1 r n n n n (1 r) Ti, (1 r) Ti, ( Ti 1, Ti 1, Ti, 1 Ti, 1 ( Ti 1, Ti 1, Ti, 1 Ti, 1 Centered in time over dt = Sum explicit and implicit RHSs (given above), divided by two ( n1 n n1 n1 n1 n1 n1 n1 i, i, i1, i, i1, i, 1 i, i, 1 c c O t O x y t x y ( ( ( ), ( ) PFJL Lecture 14, 6

27 Parabolic PDEs: Two spatial dimensions Crank-Nicolson Implicit (for Δx=Δy): 1 r r (1 r) T (1 r) T T T T T T T T T Five unknowns at the (n+1) time => penta-diagonal Either elimination procedure or iterative scheme (Jacobi/Gauss- Seidel/SOR) but not always efficient ( ( n n n n n n n n n n i, i, i 1, i 1, i, 1 i, 1 i 1, i 1, i, 1 i, 1 Alternating-Direction Implicit (ADI) schemes Provides a mean for solving parabolic PDEs with tri-diagonal matrices In D: each time increment is executed in two half steps: each step is conditionally stable, but combination of two half-steps is unconditionally stable (similar to Crank-Nicolson behavior) It is one but a group of schemes called splitting methods Extended to 3D (time increment divided in 3): varied stability properties PFJL Lecture 14, 7

28 Parabolic PDEs: Two spatial dimensions ADI scheme (Two Half steps in time) t McGraw-Hill. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Source: Chapra, S., and R. Canale. Numerical Methods for Engineers. McGraw-Hill, ) From time n to n+1/: Approximation of nd order x derivative is explicit, while the y derivative is implicit. Hence, tri-diagonal matrix to be solved: T T T T T T T T n1/ n n n n n1/ n1/ n1/ i, i, i1, i, i1, i, 1 i, i, 1 c c O x y t / x y ( ( ) ) From time n+1/ to n+1: Approximation of nd order x derivative is implicit, while the y derivative is explicit. Another tri-diagonal matrix to be solved: T T T T T T T T n1 n1/ n1 n1 n1 n1/ n1/ n1/ i, i, i1, i, i1, i, 1 i, i, 1 c c O x y t / x y ( ( ) PFJL Lecture 14, 8

29 Parabolic PDEs: Two spatial dimensions ADI scheme (Two Half steps in time) i=1 i= i=3 i=1 i= i=3 =3 = =1 y First direction Second direction x The ADI method applied along the y direction and x direction. This method only yields tridiagonal equations if applied along the implicit dimension. For Δx=Δy: Image by MIT OpenCourseWare. After Chapra, S., and R. Canale. Numerical Methods for Engineers. McGraw-Hill, ) From time n to n+1/: rt n 1/ i, 1 (1 r ) T n 1/ rt n 1/ rt n (1 r )T n rt n i, i, 1 i 1, i, i 1, ) From time n+1/ to n+1: rt n 1 i 1, n 1 (1 r ) T i, rt n 1 rt n 1/ i 1, i, 1 (1 r )T i, rt i n 1/ n 1/, 1 PFJL Lecture 17, 4

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