# Choas and instabilities in finite difference approximations to nonlinear differential equations

## Abstract

The numerical solution of time-dependent ordinary and partial differential equations by finite difference techniques is a common task in computational physics and engineering The rate equations for chemical kinetics in combustion modeling are an important example. They not only are nonlinear, but they tend to be stiff, which makes their solution a challenge for transient problems. We show that one must be very careful how such equations are solved In addition to the danger of large time-marching errors, there can be unphysical chaotic solutions that remain numerically stable for a range of time steps that depends on the particular finite difference method used We point out that the solutions of the finite difference equations converge to those of the differential equations only in the limit as the time step approaches zero for stable and consistent finite difference approximations The chaotic behavior observed for finite time steps in some nonlinear difference equations is unrelated to solutions of the differential equations, but is connected with the onset of numerical instabilities of the finite difference equations This behavior suggests that the use of the theory of chaos in nonlinear iterated maps may be useful in stability anlaysis of finite difference approximations to nonlinearmore »

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 292334

- Report Number(s):
- UCRL-ID-131333

ON: DE98058886; BR: DP0101031

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

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 1 Jul 1998

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; NONLINEAR PROBLEMS; DIFFERENTIAL EQUATIONS; FINITE DIFFERENCE METHOD; INSTABILITY; FLUID MECHANICS; CHAOS

### Citation Formats

```
Cloutman, L. D., LLNL.
```*Choas and instabilities in finite difference approximations to nonlinear differential equations*. United States: N. p., 1998.
Web. doi:10.2172/292334.

```
Cloutman, L. D., LLNL.
```*Choas and instabilities in finite difference approximations to nonlinear differential equations*. United States. doi:10.2172/292334.

```
Cloutman, L. D., LLNL. Wed .
"Choas and instabilities in finite difference approximations to nonlinear differential equations". United States. doi:10.2172/292334. https://www.osti.gov/servlets/purl/292334.
```

```
@article{osti_292334,
```

title = {Choas and instabilities in finite difference approximations to nonlinear differential equations},

author = {Cloutman, L. D., LLNL},

abstractNote = {The numerical solution of time-dependent ordinary and partial differential equations by finite difference techniques is a common task in computational physics and engineering The rate equations for chemical kinetics in combustion modeling are an important example. They not only are nonlinear, but they tend to be stiff, which makes their solution a challenge for transient problems. We show that one must be very careful how such equations are solved In addition to the danger of large time-marching errors, there can be unphysical chaotic solutions that remain numerically stable for a range of time steps that depends on the particular finite difference method used We point out that the solutions of the finite difference equations converge to those of the differential equations only in the limit as the time step approaches zero for stable and consistent finite difference approximations The chaotic behavior observed for finite time steps in some nonlinear difference equations is unrelated to solutions of the differential equations, but is connected with the onset of numerical instabilities of the finite difference equations This behavior suggests that the use of the theory of chaos in nonlinear iterated maps may be useful in stability anlaysis of finite difference approximations to nonlinear differential equations, providing more stringent time step limits than the formal linear stability analysis that tests only for unbounded solutions This observation implies that apparently stable numerical solutions of nonlinear differential equations by finite difference techniques may in fact be contaminated (if not dominated) by nonphysical chaotic parasitic solutions that degrade the accuracy of the numerical solution We demonstrate this phenomenon with some solutions of the logistic equation and a simple two-dimensional computational fluid dynamics example},

doi = {10.2172/292334},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1998},

month = {7}

}