Evaluation of Chebyshev pseudospectral methods for third order differential equations

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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(N5) 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(N4). 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.

Original languageEnglish (US)
Pages (from-to)255-281
Number of pages27
JournalNumerical Algorithms
Volume16
Issue number3-4
StatePublished - Dec 1 1997

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Keywords

  • Accuracy
  • Finite differences
  • Pseudospectral Chebyshev
  • Third order equations
  • Transformed methods

ASJC Scopus subject areas

  • Applied Mathematics

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