### Abstract

The convergence theory for numerical methods approximating time-dependent problems parallels the theory of ordinary differential equations (ODEs) where two types of behavior are studied, namely: (1) the finite time solution and (2) the long-time asymptotic behavior where the solution either passes through an initial transient state and sets into a steady state, or evolves into a periodic or chaotic motion, or escapes to infinity. We describe the notions of consistency, stability, local and global error estimates, resolution and order of accuracy, followed by Lax-Richtmyer equivalence theorem. The rest of the chapter is devoted to practical implications of the convergence theory in terms of the resolution and error estimates together with von Neumann and CFL stability restrictions.

Original language | English (US) |
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Title of host publication | Mathematics in Science and Engineering |

Publisher | Elsevier |

Pages | 109-144 |

Number of pages | 36 |

Edition | C |

DOIs | |

State | Published - Jan 1 2010 |

Externally published | Yes |

### Publication series

Name | Mathematics in Science and Engineering |
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Number | C |

Volume | 213 |

ISSN (Print) | 0076-5392 |

### Keywords

- Consistency
- Convergence
- Fourier analysis
- Global error estimate
- Local truncation error
- Order of accuracy
- Resolution
- Stability
- Von Neumann and CFL necessary stability conditions

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

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## Cite this

*Mathematics in Science and Engineering*(C ed., pp. 109-144). (Mathematics in Science and Engineering; Vol. 213, No. C). Elsevier. https://doi.org/10.1016/S0076-5392(10)21308-5