Runge-Kutta and MacCormack dynamics

Our topic is the long term simulation of nonlinear differential equations, with a focus on spurious behaviour. We outline some of the recent work that has been done for initial value ordinary differential equations, and emphasize that finite, but spurious, solutions can arise for realistic choices of the timestep. As a concrete example, we exhibit spurious fixed points arising for the Improved Euler method with a logistic right-hand side. We then consider MacCormack's finite difference method applied to a linear advection equation with source term. From a linear stability perspective, due to of a lack of convexity in the stability region, instabilities may surface if the timestep is reduced while the Courant number is fixed at a realistic level. With a nonlinear source term, MacCormack's method admits spatially uniform spurious steady states that correspond to spurious fixed points of the Improved Euler method. We examine the stability of these spurious steady states for the logistic source term, and give numerical results.