AMS527: Numerical Analysis II

Transcription

1 AMS527: Numerical Analysis II Supplementary Material on Finite Different Methods for Two-Point Boundary-Value Problems Xiangmin Jiao Stony Brook University Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II / 5

2 A Simple Finite Difference Method Consider simple second-order ODE u (x) = f (x) for 0 < x <, with some given boundary conditions u(0) = α and u() = β FDM computes grid functions of U 0, U,, U m, U m+, where U j is approximation to solution u(x j ) at jth grid point x j = jh and h = /(m + ) is mesh width Replace u (x) by finite difference operator (such as centered difference approximation) D 2 U j = h 2 (U j 2U j + U j+ ), we obtain a set of algebraic equations or h 2 (U j 2U j + U j+ ) = f (x j ) for j =, 2,, m, AU = F Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 2 / 5

3 Linear System from Simple FDM h U U 2 U 3 U m U m = f (x ) α/h 2 f (x 2 ) f (x 3 ) f (x m ) f (x m ) β/h 2 Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 3 / 5

4 Error Let Û = [u(x ), u(x 2 ),, u(x m )] be vector of true values Error vector is E = U Û, composed of errors at each grid point Error norms typically used are E = max E j j m m E = h E j j= m E 2 = h E j 2 j= E = U Û is referred to as global error Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 4 / 5

5 Local Truncation Error (LTE) Local truncation error (LTE), denoted by τ j, ie, τ j = D 2 (u) f (x j ) for j =, 2,, m For centered difference approximation, τ j = h 2 (u j 2u j + u j+ ) f (x j ) [ = u (x j ) + ] 2 h2 u (x j ) + O(h 4 ) f (x j ) = 2 h2 u (x j ) + O(h 4 ) Assume u is bounded, then τ j = O(h 2 ) as h 0 Let τ be vector composed of τ j, then τ = AÛ F, or AÛ = τ + F Therefore, AE = A(U Û) = τ Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 5 / 5

6 Stability Suppose a finite difference method for a linear BVP gives a sequence of matrix equations of form A h U h = F h, where h is the mesh width, and the superscript denotes solution on grid with mesh grading h We say the method is stable if (A h ) exists for all h sufficiently small (say h < h 0 ) and if there is a constant C, independent of h, such that (A h ) C for all h < h 0 Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 6 / 5

7 Consistency We say that a method is consistent with differential equation and boundary conditions if τ h 0 as h 0 Typically, τ h = O(h p ) for some integer p > 0, and so it is consistent Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 7 / 5

8 Convergence A method is said to be convergent if E h 0 as h 0 In general, This is because consitency + stability convergence E h (A h ) τ h C τ h 0 as h 0 This is known as fundamental theorem of finite difference methods This is analogous to Lax equivalence theorem for finite difference methods for the numerical solution of partial differential equations: For a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 8 / 5

9 Stability in 2-Norm Challenge in proof of convergence often lies in verification of stability (instead of consistency) In 2-norm, for symmetric matrices A 2 = ρ(a) = max ρ m λ p and ( ) A 2 = ρ(a ) = min λ p ρ m Therefore, one needs to show eigenvalues of A is bounded away from zero as h 0 Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 9 / 5

10 Stability of Model Problem The m eigenvalues of A are λ p = 2 (cos(pπh) ) for p =, 2,, m, h2 and their corresponding eigenvector u p has jth component u p j = sin(pπjh) Smallest eigenvalue of A (in magnitude) is λ = 2 (cos(πh) ) h2 = 2 ( h 2 2 π2 h 2 + ) 24 π4 h 4 + O(h 6 ) = π 2 + O(h 2 ), which is bounded away from 0 as h 0 Therefore, E h 2 τ h π 2 2, so E 2 = O(h 2 ) It can also be shown that E = O(h 2 ), but it is more involved Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 0 / 5

11 Neumann Boundary Conditions Consider simple second-order ODE with boundary conditions u (x) = f (x) for 0 < x <, u (0) = σ, u() = β Discretize interval [0, ] by introducing m interior points such that 0 = x 0 < x < x 2 < < x m < x m < x m+ = First Attempt: Approximate u (0) = 0 using one-sided derivative U U 0 = σ h Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II / 5

12 First Attempt Then equation becomes h h h 2 0 h 2 Local truncation error is O(h), because U 0 U U 2 U m U m+ τ 0 = h 2 (hu(x ) hu(x 0 )) σ = u (x 0 ) + 2 hu (x 0 ) + O(h 2 ) σ = 2 hu (x 0 ) + O(h 2 ), This translates into a global error of O(h) = σ f (x ) f (x 2 ) f (x m ) β Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 2 / 5

13 First Second-Order Accurate Approach Use centered approximation to u (0) = σ by introducing U h 2 (U 2U 0 + U ) = f (x 0 ) 2h (U U ) = σ Eliminate U from the two equations and we obtain We then have h h h 2 h ( U 0 + U ) = σ + h 2 f (x 0) 0 h 2 U 0 U U 2 U m U m+ = σ + h 2 f (x 0) f (x ) f (x 2 ) f (x m ) β Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 3 / 5

14 Second Second-Order Accurate Approach Use second-order one-sided approximation to u (0) = σ ( 3 h 2 U 0 2U + ) 2 U 2 = σ We then have 3 2 h 2h h 2 2 h 0 h 2 U 0 U U 2 U m U m+ = σ f (x ) f (x 2 ) f (x m ) β This approach increases bandwidth of matrix, but is more general and easier to generalize to higher order accuracy and to nonuniform grids Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 4 / 5

15 References Randall J LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007 Xiangmin Jiao Stony Brook University AMS527: Numerical Analysis II 5 / 5