Modified equation techniques for reaction-diffusive systems. part 2: time-linearization and operator-splitting methods

Abstract

Modified equation methods are applied with time-linearization and operator-splitting techniques to the study of a reaction-diffusion equation which has an exact travelling wave solution. Comparisons between the exact and computed solutions have been made in terms of L 2- norm errors and wave speeds. It is shown that modified time-linearization methods are less accurate than time-linearized techniques if the time derivatives are evaluated by means of first-order accurate approximations. However, if second-order accurate discretizations are used to evaluate the time derivatives, modified time-linearization methods are more accurate than the Beam-Warming and Briley-McDonald schemes. Second-order accurate (in time) modified time-linearization methods yield L 2- norm errors which are almost independent of time and of the time step used in the calculations. The accuracy of modified time-linearization techniques deteriorates as both the time step and grid spacing are increased. Second-order accurate (in time) modified time-linearization algorithms are more efficient for the same accuracy than the Beam-Warming technique. Modified operator-splitting methods yield wave speeds and L 2- norm errors which are almost independent of the explicit or implicit treatment of the appended terms. A first-order accurate (in time) operator-splitting method was found to yield more accurate wave speeds than modified operator-splitting techniques indicating that no accuracy improvements are achieved by employing modified equation schemes in operator-splitting procedures.