Numerical Wave Propagation in an Advection Equation with a Nonlinear Source Term

Abstract

The Cauchy and initial boundary value problems are studied for a linear advection equation with a nonlinear source term. The source term is chosen to have two equilibrium states, one unstable and the other stable as solutions of the underlying characteristic equation. The true solutions exhibit travelling waves which propagate from one equilibrium to another. The speed of propagation is dependent on the rate of decay of the initial data at infinity A class of monotone explicit finite-difference schemes are proposed and analysed; the schemes are upwind in space for the advection term with some freedom of choice for the evaluation of the nonlinear source term. Convergence of the schemes is demonstrated and the existence of numerical waves, mimicking the travelling waves in the underlying equation, is proved. The convergence of the numerical wave-speeds to the true wave-speeds is also established. The behaviour of the scheme is studied when the monotonicity criteria are violated due to stiff source terms, and oscillations and divergence are shown to occur. The behaviour is contrasted with a split-step scheme where the solution remains monotone and bounded but where incorrect speeds of propagation are observed as the stiffness of the problem increases.