Some results on the stability and dynamics of finite difference approximations to nonlinear partial differential equations

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.