# A note on the stability and accuracy of finite difference approximations to differential equations

## Abstract

There are many finite difference approximations to ordinary and partial differential equations, and these vary in their accuracy and stability properties. We examine selected commonly used methods and illustrate their stability and accuracy using both linear stability analysis and numerical examples. We find that the formal order of accuracy alone gives an incomplete picture of the accuracy of the method. Specifically, the Adams-Bashforth and Crank-Nicholson methods are shown to have some undesirable features for both ordinary and partial differential equations.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab., CA (United States)

- Sponsoring Org.:
- USDOE Office of Energy Research, Washington, DC (United States)

- OSTI Identifier:
- 420369

- Report Number(s):
- UCRL-ID-125549

ON: DE97050754; TRN: 97:000757

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: Sep 1996

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFERENTIAL EQUATIONS; FINITE DIFFERENCE METHOD; WAVE EQUATIONS; DIFFUSION; STABILITY

### Citation Formats

```
Cloutman, L.D.
```*A note on the stability and accuracy of finite difference approximations to differential equations*. United States: N. p., 1996.
Web. doi:10.2172/420369.

```
Cloutman, L.D.
```*A note on the stability and accuracy of finite difference approximations to differential equations*. United States. doi:10.2172/420369.

```
Cloutman, L.D. Sun .
"A note on the stability and accuracy of finite difference approximations to differential equations". United States. doi:10.2172/420369. https://www.osti.gov/servlets/purl/420369.
```

```
@article{osti_420369,
```

title = {A note on the stability and accuracy of finite difference approximations to differential equations},

author = {Cloutman, L.D.},

abstractNote = {There are many finite difference approximations to ordinary and partial differential equations, and these vary in their accuracy and stability properties. We examine selected commonly used methods and illustrate their stability and accuracy using both linear stability analysis and numerical examples. We find that the formal order of accuracy alone gives an incomplete picture of the accuracy of the method. Specifically, the Adams-Bashforth and Crank-Nicholson methods are shown to have some undesirable features for both ordinary and partial differential equations.},

doi = {10.2172/420369},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1996},

month = {9}

}

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