### 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.

Original language | English (US) |
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Pages (from-to) | 255-281 |

Number of pages | 27 |

Journal | Numerical Algorithms |

Volume | 16 |

Issue number | 3-4 |

State | Published - Dec 1 1997 |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Numerical Algorithms*,

*16*(3-4), 255-281.