### 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) |
---|---|

Pages (from-to) | 255-281 |

Number of pages | 27 |

Journal | Numerical Algorithms |

Volume | 16 |

Issue number | 3-4 |

State | Published - 1997 |

### Fingerprint

### 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.

**Evaluation of Chebyshev pseudospectral methods for third order differential equations.** / Renaut, Rosemary; Su, Yi.

Research output: Contribution to journal › Article

*Numerical Algorithms*, vol. 16, no. 3-4, pp. 255-281.

}

TY - JOUR

T1 - Evaluation of Chebyshev pseudospectral methods for third order differential equations

AU - Renaut, Rosemary

AU - Su, Yi

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

KW - Accuracy

KW - Finite differences

KW - Pseudospectral Chebyshev

KW - Third order equations

KW - Transformed methods

UR - http://www.scopus.com/inward/record.url?scp=0043234049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0043234049&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0043234049

VL - 16

SP - 255

EP - 281

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

IS - 3-4

ER -