Title

Some Results On The Stability And Dynamics Of Finite Difference Approximations To Nonlinear Partial Differential Equations

Abstract

A miscellany of results on the nonlinear instability and dynamics of finite difference discretizations of the Burgers and Kortweg de Vries equations is obtained using a variety of phase‐plane, functional analytic, and regularity methods. For the semidiscrete (space‐discrete, time‐continuous) schemes, large‐wave‐numer instabilities occurring in special exact solutions are investigated, and parameter values for which the semidiscrete scheme is monotone are considered. For fully discrete schemes (space and time discrete), large‐wave‐number instabilities introduced by various time‐stepping schemes such as forward Euler, leapfrog, and Runge–Kutta schemes are analyzed. Also, a time step restriction for the monotonicity of the forward‐Euler time‐stepping scheme, and regularity of a 4‐stage monotone/conservative Runge–Kutta time stepping are investigated. The techniques used here may be employed, in conjunction with bifurcation‐theoretic and weakly nonlinear analyses, to analyze the stability of numerical schemes for other nonlinear partial differential equations of both dissipative and dispersive varieties. © 1993 John Wiley & Sons, Inc. Copyright © 1993 Wiley Periodicals, Inc.

Publication Date

1-1-1993

Publication Title

Numerical Methods for Partial Differential Equations

Volume

9

Issue

2

Number of Pages

117-133

Document Type

Article

Identifier

scopus

Personal Identifier

scopus

DOI Link

https://doi.org/10.1002/num.1690090203

Socpus ID

0027558621 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0027558621

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