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Burden - Numerical Analysis 5e (PWS, 1993) Episode 3 Part 7

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 3 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 12.2 Parabolic Partial-Differential Equations 643 This method does have local truncation error of order Off -T- ĩ2 but unfortunately it also has serious stability problems see Exercise 6 . A more rewarding method is derived by averaging the Forward-Difference method at the jth step in t WiJ l - - 2wu _ n ------ a - u k-------------------------------------------------h2- which has local truncation error k d2u ọ rF hO 2 dt and the Backward-Difference method at the j -T- l st Stepan t i j i - i j _ 2WỈ 1 - 1 - 2w WZ_1J 1 ------ a ----------------A------------- 0. k h2 which has local truncation error fc d2u rB - - Zĩffị Aj Offz . 2 ỔÍ If we assume that 02w d2u Aj at at then the averaged difference method - i j CT H u - 2wfJ WÍ 1J 1 - 2-Wij k 2 _ h2 h2 J has local truncation error of order Off2 A A2 provided of course that the usual differentiability conditions are satisfied. This is known as the Crank-Nicolson method and is represented in the matrix form 12.15 Ấwơ 1 - Bv j for each 0 1 2 where À wư w J w2 . jY h J and the matrices A and B are given by 1 A 0--.0 _À _ 2 --_ ỉ 0 . ị A A ị -0 A 2 0. 0 l Ả 644 CHAPTER 12 Numerical Solutions to Partial-Differential Equations and 1 A 0- -. 0 A 2 0. . 0 A -. 2 0. 0 I 1 - A Since A is a positive definite sưictly diagonally dominant and tridiagonal matrix it is nonsingular. Either the Crout Factorization for Tridiagonal Linear System Algorithm 6.7 or the SOR Algorithm 7.3 can be used to obtain wc 1 from w for each j 0 1 2.Algorithm 12.3 incorporates Crout factorization into the Crank-Nicolson technique. As in Algorithm 12.2 a finite length for the time interval must be specified to determine a stopping procedure. ALGORITHM 12.3 Crank- ỉicolson To approximate the solution to the parabolic partial-differential equation du J o d2u _ _ -7- x t a2 7-7 x i 0 0 X I 0 t T ộ ỠX subject to the boundary conditions m 0 í u l f 0 0 t T and the initial conditions M. x 0 x 0 X Ỉ INPUT endpoint Ỉ maximum time T constant a integers m 3 N 1. .