Numerical Solutions to Partial Differential Equations

Transcription

1 Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University

2 A Model Problem in a 2D Box Region Let us consider a model problem of parabolic equation: u t = a u xx + u yy, x, y Ω, t > 0, ux, y, 0 = u 0 x, y, 0, x, y Ω, ux, y, t = 0, x, y Ω, t > 0, where a > 0 is a constant, Ω = 0, X 0, Y R 2. 1 For integers N x 1 and N y 1, let h x = x = XNx 1 and h y = y = YNy 1 be the grid sizes in the x and y directions; 2 a uniform parallelopiped grid with the set of grid nodes J Ω R+ = {x j, y k, t m : 0 j N x, 0 k N y, m 0}, where x j = j h x, y k = k h y, t m = mτ τ > 0 time step size. 3 the space of grid functions U = {U m = Ux j, y k, t m : 0 j N x, 0 k N y, m 0}.

3 Explicit and Implicit Schemes in Box Regions The Explicit Scheme in a 2D Box Region The Forward Explicit Scheme in a 2D Box Region The forward explicit scheme and its error equation U m+1 U m [ U m j+1,k 2U m = a + Um j 1,k τ hx 2 + Um +1 2Um + ] Um 1 hy 2. e m+1 = [1 2µ x + µ y ] e+µ m x e m j+1,k + ej 1,k m +µy e m +1 + e 1 m T m where µ x = a τ, µ hx 2 y = a τ hy 2 are the grid ratios in x and y directions. The truncation error is Tux, y, t = 1 2 u ttx, y, tτ a 4 12 x ux, y, thx 2 + y 4 ux, y, thy 2 + Oτ 2 + hx 4 + hy 4. τ, 3 / 35

4 Explicit and Implicit Schemes in Box Regions The Explicit Scheme in a 2D Box Region The Forward Explicit Scheme in a 2D Box Region 1 the condition for the maximum principle: µ x + µ y 1 2 ; 2 Fourier modes and amplification factors λ l l = l x, l y : U m = λm l e iαx x j +α y y k = λ m l e ilx jπn 1 x +l y kπny 1, α x = lx π X, N x + 1 l x N x, α y = ly π Y, N y + 1 l y N y ; l x, l y represent the frequency or wave number in x, y direction. [ ] 3 2 αx hx amplification factor λ l = 1 4 µ x sin 2 + µ y sin 2 α y h y 2 [ = 1 4 µ x sin 2 lx π 2N x + µ y sin 2 l y π 2N x ]; 4 L 2 stable if and only if µ x + µ y 1/2; 5 Convergence rate is Oτ + h 2 x + h 2 y. 4 / 35

5 The θ-scheme in a 2D Box Region [ ] [ ] U m+1 U m δx 2 = 1 θa τ hx 2 + δ2 y hy 2 U m δx 2 + θa hx 2 + δ2 y hy 2 [ U m j+1,k 2U m = 1 θa + Um j 1,k + Um +1 2Um + ] Um 1 + θa h 2 x [ U m+1 j+1,k 2Um+1 + U m+1 j 1,k hx 2 + Um+1 The error equation: [1 + 2θµ x + µ y ] e m+1 h 2 y +1 2Um+1 + U m+1 h 2 y 1 U m+1 ] = θ [µ x e m+1 j+1,k + em+1 j 1,k + µ y e m em [1 21 θµ x + µ y ] e m + 1 θ [ µ x e m j+1,k + e m j 1,k + µy e m +1 + e m 1] τt m+ 1 2, ].

6 Explicit and Implicit Schemes in Box Regions The θ-scheme in a 2D Box Region The θ-scheme in a 2D Box Region The truncation error T m+ j = { Oτ 2 + h 2 x + h 2 y, if θ = 1 2, Oτ + h 2 x + h 2 y, if θ 1 2, 1 the condition for the maximum principle: 2µ x + µ y 1 θ 1; 2 Fourier modes: U m = λm l e iαx x j +α y y k = λ m l e l = l x, l y, α x = lx π X, αy = ly π Y, αx x j = 3 amplification factor 1 41 θ λ l = 1 + 4θ lx jπ Nx, αy y ly kπ k = Ny ; [ µ x sin 2 lx π 2N x [ µ x sin 2 lx π 2N x lx jπ ly kπ i + Nx Ny, ] + µ y sin 2 l y π 2N x + µ y sin 2 l y π 2N x ] ; 6 / 35

7 Explicit and Implicit Schemes in Box Regions The θ-scheme in a 2D Box Region The θ-scheme in a 2D Box Region 4 for θ 1/2, unconditionally L 2 stable; 5 for 0 θ < 1/2, L 2 stable iff 21 2θµ x + µ y 1; 6 the matrix of the linear system is still symmetric positive definite and diagonal dominant, however, each row has now up to 5 nonzero elements with a band width of the order Oh 1 ; 7 if solved by the Thompson method, the cost is Oh 1 times of that of the explicit scheme; 8 in 3D, µ x + µ y µ x + µ y + µ z, and Oh 1 Oh 2, if solved by the Thompson method, the cost is Oh 2 times of that of the explicit scheme. 7 / 35

8 2D and 3D ADI and LOD Schemes Alternative Approaches for Solving n-d Parabolic Equations To reduce the computational cost, we may consider to apply highly efficient iterative methods to solve the linear algebraic equations, for example, the preconditioned conjugate gradient method; the multi-grid method; etc.. Alternatively, to avoid the shortcoming of the implicit difference schemes for high space dimensions, we may develop the alternating direction implicit ADI schemes; the locally one dimensional LOD schemes. 8 / 35

9 2D and 3D ADI and LOD Schemes A 2D Alternating Direction Implicit ADI Scheme A Fractional Steps 2D ADI Scheme by Peaceman and Rachford 1 in the odd fractional steps implicit in x and explicit in y, and in even fractional steps implicit in y and explicit in x: µ xδx 2 U m+ 1 2 = µ y δy 2 U, m µ y δ 2 y U m+1 = µ xδ 2 x U m+ 1 2, 2 numerical boundary conditions are easily imposed directly by those of the original problem, since U m+ 1 2 u m+ 1 2 ; 9 / 35

10 2D and 3D ADI and LOD Schemes A 2D Alternating Direction Implicit ADI Scheme A Fractional Steps 2D ADI Scheme by Peaceman and Rachford 3 one step equivalent scheme µ xδx µ y δy 2 U m+1 = µ xδx µ y δy 2 U. m 4 Crank-Nicolson scheme µ xδx µ y δy 2 U m+1 = µ xδx µ y δy 2 U. m 5 Since µ x µ y δ 2 xδ 2 y δ t u m+ 1 2 = a 2 τ 3 [u xxyyt ] m Oτ 5 + τ 3 h 2 x + h 2 y, the truncation error of the scheme is Oτ 2 + h 2 x + h 2 y. 10 / 35

11 2D and 3D ADI and LOD Schemes A 2D Alternating Direction Implicit ADI Scheme Stability of the 2D ADI Scheme by Peaceman and Rachford 1 For the Fourier mode U m = λm l e ilx jπn 1 x +l y kπn 1 y, λ l = 1 2 µ x sin 2 lx π 2N x 1 2 µ y sin 2 ly π 2N y µ x sin 2 lx π 2N x µ y sin 2 ly π 2N y, the scheme is unconditionally L 2 stable; 2 the two fractional steps can be equivalently written as 1 + µ xu m+ 1 2 = 1 µ y U m + µy U m 1 + U m µ x µ y U m+1 = 1 µ xu m µx U m+ 1 2 j 1,k Um+ 2 j+1,k + µy 2 thus, the maximum principle holds if max{µ x, µ y } 1; U m+ 1 2 j 1,k + Um+ 1 2 j+1,k, U m U m+1 +1, 11 / 35

12 2D and 3D ADI and LOD Schemes A 2D Alternating Direction Implicit ADI Scheme Cost of the 2D ADI Scheme by Peaceman and Rachford 1 order the linear system by N x 1 k + j and N y 1 j + k in odd and even steps, the corresponding matrixes are tridiagonal; 2 the computational cost: 3 times of that of the explicit scheme. 12 / 35

13 2D and 3D ADI and LOD Schemes A 2D Alternating Direction Implicit ADI Scheme The Idea of Peaceman and Rachford Doesn t Work for 3D In 3D, we can still construct fractional step scheme by 1 dividing each time step into 3 fractional steps; 2 introducing U m+ 1 3 and U m+ 2 3 at t m+ 1 3 and t m+ 2 ; 3 3 in the 3 fractional time steps, applying schemes which are in turn implicit in x-, y- and z-direction and explicit in the other 2 directions respectively. However, the equivalent one step scheme is, up to a higher order term, the same as the θ-scheme with θ = 1/3, which has a local truncation error Oτ + h 2 x + h 2 y instead of what we expect to have Oτ 2 + h 2 x + h 2 y for an ADI method. 13 / 35

14 The Key Properties that Make an ADI Scheme Successful 1 Implicit only in one dimension in each fractional time step, thus, if properly ordered, the matrix of the linear algebraic equations is tridiagonal as well as diagonally dominant, hence the computational cost is significantly reduced. 2 In the equivalent one step scheme, the implicit part is the product of 1D implicit difference operators, the explicit part is the product of 1D explicit difference operators in a half time step, + certain h.o.t. small perturbations, which guarantees the scheme is unconditionally L 2 stable; 3 The difference between the equivalent one step scheme and the Crank-Nicolson scheme is a higher order term, which guarantees that the local truncation error is Oτ 2 + h 2.

15 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes Target Equivalent One Step Schemes of ADI Methods For the 3D model problem, we aim to develop ADI finite difference schemes which have a one time step equivalent scheme of the form µ xδx µ y δy µ zδz 2 U m+1 = µ xδx µ y δy µ zδz 2 U m, or µ xδx µ y δy µ zδz 2 = µ xδx µ y δy 2 U m µ zδ 2 z U m + h.o.t. 15 / 35

16 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes An Extendable ADI Scheme Proposed by D yakonov For the 3D model problem, the ADI Scheme of D yakonov is 1 µ x 2 δ2 x U m+ = 1 + µ x 2 δ2 x 1 + µ y 2 δ2 y 1 + µ z 2 δ2 z U m, 1 µ y 2 δ2 y U m+ = U m+, 1 µ z 2 δ2 z U m+1 = U m+. The scheme obviously has all the key properties. However, there is one thing need to be taken care of: U m+, U m+ are not approximate solutions, their boundary conditions may not be directly derived from that of the original problem. 16 / 35

17 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes Numerical Boundary Conditions for U m+, U m+ Since U m+1 u m+1, Um+1 = u m+1 on the boundary j = 0, N x, or k = 0, N y, or l = 0, N z. Thus, the boundary condition { U m+ } : j = 0, N x, 0 k N y, 0 < l < N z, or k = 0, N y ; 0 j N x, 0 < l < N z can be obtained from U m+ = 1 µz 2 δ2 z U m+1. The boundary { condition } U m+ : j = 0, N x, 0 < k < N y, 0 < l < N z. can be obtained from U m+ = 1 µy 2 δ2 y U m+ and the boundary values for U m+ on the nodes j = 0, N x. 17 / 35

18 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes An Extendable ADI Scheme Proposed by Douglas and Rachford For the 3D model problem, the Scheme is of the form U m+1 = U m µ xδx 2 U m+1 + U m + µ y δy 2 U m + µ zδz 2 U m, U m+1 = U m µ y δy 2 U m+1 + U m µ y δy 2 U m, U m+1 = U m µ zδz 2 U m+1 +U m µ z δz 2 U m. Since U m+1 and U m+1 are approximate solutions at t m+1, their boundary conditions can be directly derived from that of the original problem.the scheme has all the key properties with the h.o.t.= 1 4 µ xµ y µ z δ 2 xδ 2 y δ 2 z U m. 18 / 35

19 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes The Idea of the ADI Scheme of Douglas and Rachford The construction of the scheme may be viewed as a predictioncorrection process, which may also be explained as follows: In the 1st fractional step, Crank-Nicolson to x, explicit to y, z. In the 2nd fractional step, to improve stability and accuracy in y, the y-direction is corrected by the Crank-Nicolson scheme: U m+1 = U m µ xδx 2 U m+1 + U m µ y δy 2 U m+1 + U m + µ z δz 2 U m. Note, this minus the 1st equation gives the second in the scheme. 19 / 35

20 2D and 3D ADI and LOD Schemes 3D Alternate Direction Implicit ADI Schemes The Idea of the ADI Scheme of Douglas and Rachford In the 3rd, z-direction is corrected by the Crank-Nicolson scheme: U m+1 = U m µ xδ 2 x µ y δ 2 y U m+1 U m+1 + U m + U m µ zδz 2 U m+1 + U m. Note, the 3rd equation in the scheme is simply the difference of the above two equations. Since U m+1 and U m+1 are approximations of U m+1, numerical boundary conditions can be directly derived from those of the original problem. 20 / 35

21 2D and 3D ADI and LOD Schemes 3D Locally One Dimensional LOD Schemes An Extendable Locally One Dimensional LOD Scheme 1 In the ith fractional step, the problem is treated as a one dimensional problem of the ith dimension. 2 1D Crank-Nicolson scheme is applied to each fractional step. 3 For the 3D model problem, the LOD scheme is given as µ xδx 2 U m+ = µ xδx 2 U, m µ y δy 2 U m+ = µ y δy 2 U m+, µ zδ 2 z U m+1 = µ zδ 2 z 4 The scheme has all of the key properties. U m+. 21 / 35

22 2D and 3D ADI and LOD Schemes 3D Locally One Dimensional LOD Schemes Impose Boundary Conditions for the LOD Scheme However, U m+, U m+ are nonphysical, their boundary conditions must be handled with special care. 1 multiplying µ zδz 2 on the 3rd equation leads to µ2 zδz 4 U m+ = 1 µ z δz µ2 zδz 4 U m+1, Hence, by omitting the Oτ 2 order terms, the boundary condition for U m+ can be given by: U m+ = 1 µ z δz 2 U m+1, k = 0, N y, 0 j N x, 0 < l < N z, j = 0, N x, 0 k N y, 0 < l < N z. 22 / 35

23 2D and 3D ADI and LOD Schemes 3D Locally One Dimensional LOD Schemes Impose Boundary Conditions for the LOD Scheme 2 Similarly, multiplying µ y δ 2 y on the 2nd equation, omitting the Oτ 2 and higher order terms, the boundary condition for U m+ can be given by: U m+ = 1 µ y δy 2 U m+, j = 0, N x, 0 < k < N y, 0 < l < N z. 23 / 35

24 FDM for General Parabolic Problems in High Dimensions General Domain and Boundary Conditions General Domain and Boundary Conditions 1 Approximate boundary conditions can be established with the similar methods used in for elliptic problems. 2 Approximate boundary conditions can further restrict the stability condition for explicit schemes, thus implicit schemes are even more preferable. 3 ADI and LOD schemes are in more complicated forms. 4 Some special regions with non-planar boundaries can be transformed into box regions by curvilinear coordinate systems, say, the polar coordinate system can be used for a circular region; and the cylindrical coordinate system can be used for cylindrical regions, etc / 35

25 FDM for General Parabolic Problems in High Dimensions Variable-coefficient Equations Variable-coefficient and nonlinear Equations 1 ADI and LOD schemes can also be extended to high dimensional variable-coefficient linear, and even certain nonlinear problems. 2 In such cases, the difference operators 1 ± 1 2 µ xδx 2 and 1 ± 1 2 µ y δy 2 are generally noncommutative, i.e. 1 ± 1 2 µ xδx 2 1 ± 1 2 µ y δy 2 1 ± 1 2 µ y δy 2 1 ± 1 2 µ xδx 2, and this will introduce the so called splitting error. 3 Boundary conditions are more difficult to handle. 25 / 35

26 FDM for General Parabolic Problems in High Dimensions Variable-coefficient Equations Implicit, Semi-implicit Schemes For nonlinear problems, nonlinear algebraic equations derived from implicit schemes need to be solved with iterative methods, such as semi-implicit schemes. The idea is to apply an implicit scheme only to the principal linear part of the nonlinear equation; approximate the residual nonlinear part by an explicit scheme. 26 / 35

27 FDM for General Parabolic Problems in High Dimensions Variable-coefficient Equations Implicit, Semi-implicit Schemes and Elliptic Solver To apply the implicit schemes to linear parabolic problems and the semi-implicit schemes to nonlinear parabolic problems, it is a sequence of linear algebraic systems corresponding to certain elliptic problems we actually need to solve in the end. So, Fast solvers for elliptic problems play important roles in solving parabolic problems. 27 / 35

28 FDM for General Parabolic Problems in High Dimensions Variable-coefficient Equations Asymptotic Analysis and Extrapolation Methods The asymptotic analysis and extrapolation methods introduced in 1.5 for elliptic problems can also be extended to analyze the finite difference approximation error for parabolic problems. In particular, the error bounds can be estimated by the numerical solutions obtained on grids with different mesh sizes. For example, for the explicit scheme of the heat equation, let τ = µh 2, µ 1/2, then, we have U m hj = um hj + Oh2 + Oh 4, U 1m j 4U4m h/22j U m hj 3 = u m hj + Oh4. 28 / 35

29 Finite Difference Method for Hyperbolic equations Introduction to Hyperbolic Equations n-d 1st Order Linear Hyperbolic Partial Differential Equation: n 1 Scalar case u R 1, u t + a i u xi + b u = ψ 0, i=1 where a i, b and ψ 0 are real functions x = x 1,..., x n and t. 2 Vector case u = u 1,, u p T R p, n u t + A i u xi + B u = ψ 0, i=1 where A i, B R p p, ψ 0 R p are real functions of t and x = x 1,..., x n, and α R n, Ax, t = n i=1 α ia i x, t is real diagonalizable, i.e. Ax, t has p linearly independent eigenvectors corresponding to real eigenvalues. 3 If Ax, t = n i=1 α ia i x, t has p mutually different real eigenvalues, the system is called strictly hyperbolic.

30 Finite Difference Method for Hyperbolic equations Introduction to Hyperbolic Equations n-d 2nd Order Linear Hyperbolic Partial Differential Equation A general 2nd order scalar equation u R 1, n n n u tt + 2 a i u xi t + b 0 u t a ij u xi x j + b i u xi + cu = ψ 0, i=1 i,j=1 where a i, a ij, b i, c and ψ 0 are real functions x = x 1,..., x n and t, a ij is real symmetric positive definite. Define v = u, v 0 = u t, v i = u xi, then the above 2nd order scalar equation transforms into a first order linear system of partial differential equations for v = v, v 0, v 1,, v n T n Av t + A i v xi + Bv = ψ 0, i=1 i=1

31 Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 2nd Order Scalar Transforms to n-d 1st Order System p = n a i a 1i a ni A = 0 0 a 11 a 1n, A i = 0 a 1i 0 0, a n1 a nn 0 a ni c b 0 b 1 b n ψ 0 B = , ψ 0 = / 35

32 Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 2nd Order Scalar Transforms to n-d 1st Order System p = n Let R T AR = I A is real symmetric positive definite; 2 Introduce a new variable w = R 1 v; 3 Denote Âi = R T A i R, ˆB = R T BR, ˆψ 0 = R T ψ; 4 Âx, t = n i=1 α iâix, t is a real symmetric matrix and thus is real diagonizable for all real α i, i = 1,..., n; 5 The 2nd order scalar equation now transforms into a 1st order linear hyperbolic system of partial differential equations for w R n+2 : n w t + Â i w xi + ˆBw = ˆψ 0. i=1 32 / 35

33 Introduction to Hyperbolic Equations The Hyperbolic Equations Standard Form of n-d 1st Order Linear Hyperbolic Equations 1 The standard form of 1st order linear hyperbolic equation: n u t + a i u xi = ψ, ψ = ψ 0 b u. i=1 2 The standard form of 1st order linear hyperbolic system: n u t + A i u xi = ψ, ψ = ψ 0 B u; i=1 3 The equation system is said to be homogeneous, if ψ = 0; 33 / 35

34 Introduction to Hyperbolic Equations The Hyperbolic Equations Standard Form of n-d 1st Order Linear Hyperbolic Equations 4 In general, a higher order linear hyperbolic equation system of equations can always be transformed into a first order linear hyperbolic system of equations. 5 An equation system is said to be nonlinear, if at least one of the coefficients depends on the unknown or the right hand side term is a nonlinear function of the unknown. 34 / 35

35 Thank You! SK 2µ20, 21