MacCormack's method for advection-reaction equations

MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic problems. Here, we consider MacCormack's method applied to the linear advection equation with nonlinear source term. Various features of the method are analysed. First, we show that the conventional implementation is not stable for Courant numbers close to one unless a small time-step is used. A simple modification, based on source term averaging, is shown to remove this defect. We then examine spurious fixed points that are inherited from the underlying Runge--Kutta method. Next we consider adapting the timestep as a means of improving the efficiency of the method. Theoretical analysis based on the method of modified equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be refined. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modified error measure that takes account of immanent spatial errors.