Evaluation of Chebyshev pseudospectral methods for third order differential equations |
| |
Authors: | Renaut Rosemary Su Yi |
| |
Institution: | (1) Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA;(2) Department of Mathematics, University of Illinois, Urbana, IL 62801, USA |
| |
Abstract: | When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary
condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and
extreme eigenvalue already reaching O(N
5 for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained
by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer 3]. By a correct choice of mapping
and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N
4. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial
differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison
verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the
pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach.
This revised version was published online in August 2006 with corrections to the Cover Date. |
| |
Keywords: | pseudospectral Chebyshev third order equations finite differences transformed methods accuracy 65L05 |
本文献已被 SpringerLink 等数据库收录! |
|