### Abstract

Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the space-time discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand.

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

Pages (from-to) | 59-108 |

Number of pages | 50 |

Journal | Mathematics in Science and Engineering |

Volume | 213 |

Issue number | C |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

- Alternating-Direction-Implicit method
- Compact finite-differences
- Finite-differences
- Lagrangian interpolation
- Method of weighted residuals
- Spectral differentiation

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Mathematics in Science and Engineering*,

*213*(C), 59-108. https://doi.org/10.1016/S0076-5392(10)21307-3

**Discretization methods.** / Brio, Moysey; Webb, G. M.; Zakharian, A. R.

Research output: Contribution to journal › Article

*Mathematics in Science and Engineering*, vol. 213, no. C, pp. 59-108. https://doi.org/10.1016/S0076-5392(10)21307-3

}

TY - JOUR

T1 - Discretization methods

AU - Brio, Moysey

AU - Webb, G. M.

AU - Zakharian, A. R.

PY - 2008

Y1 - 2008

N2 - Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the space-time discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand.

AB - Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the space-time discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand.

KW - Alternating-Direction-Implicit method

KW - Compact finite-differences

KW - Finite-differences

KW - Lagrangian interpolation

KW - Method of weighted residuals

KW - Spectral differentiation

UR - http://www.scopus.com/inward/record.url?scp=77955259821&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955259821&partnerID=8YFLogxK

U2 - 10.1016/S0076-5392(10)21307-3

DO - 10.1016/S0076-5392(10)21307-3

M3 - Article

AN - SCOPUS:77955259821

VL - 213

SP - 59

EP - 108

JO - Mathematics in Science and Engineering

JF - Mathematics in Science and Engineering

SN - 0076-5392

IS - C

ER -