### Abstract

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation, u_{tt} =c^{2}u_{xx}, given by {Mathematical expression} It has already been proved that the maximal order of accuracy p of such schemes is given by p ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Padé schemes, with S = 0, s > 0, and s = 0, S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.

Original language | English (US) |
---|---|

Pages (from-to) | 83-115 |

Number of pages | 33 |

Journal | BIT Numerical Mathematics |

Volume | 35 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1995 |

### Fingerprint

### Keywords

- accuracy
- finite difference methods
- order stars
- Padé approximants
- Riemann surface
- stability
- wave equation

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics
- Software
- Computer Graphics and Computer-Aided Design

### Cite this

*BIT Numerical Mathematics*,

*35*(1), 83-115. https://doi.org/10.1007/BF01732980

**The maximal accuracy of stable difference schemes for the wave equation.** / Jeltsch, Rolf; Renaut, Rosemary; Smit, Kosie J H.

Research output: Contribution to journal › Article

*BIT Numerical Mathematics*, vol. 35, no. 1, pp. 83-115. https://doi.org/10.1007/BF01732980

}

TY - JOUR

T1 - The maximal accuracy of stable difference schemes for the wave equation

AU - Jeltsch, Rolf

AU - Renaut, Rosemary

AU - Smit, Kosie J H

PY - 1995/3

Y1 - 1995/3

N2 - We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation, utt =c2uxx, given by {Mathematical expression} It has already been proved that the maximal order of accuracy p of such schemes is given by p ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Padé schemes, with S = 0, s > 0, and s = 0, S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.

AB - We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation, utt =c2uxx, given by {Mathematical expression} It has already been proved that the maximal order of accuracy p of such schemes is given by p ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Padé schemes, with S = 0, s > 0, and s = 0, S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.

KW - accuracy

KW - finite difference methods

KW - order stars

KW - Padé approximants

KW - Riemann surface

KW - stability

KW - wave equation

UR - http://www.scopus.com/inward/record.url?scp=34249754431&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249754431&partnerID=8YFLogxK

U2 - 10.1007/BF01732980

DO - 10.1007/BF01732980

M3 - Article

VL - 35

SP - 83

EP - 115

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 1

ER -