Maccormack's Method for Advection-reaction Equations

MacCormack's method is an explicit, second order nite diierence 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 modiication, based on source term averaging, is shown to remove this defect. We then examine spurious xed points that are inherited from the underlying Runge{Kutta method. Next we consider adapting the time-step as a means of improving the eeciency of the method. Theoretical analysis based on the method of modiied equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be reened. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modiied error measure that takes account of immanent spatial errors.