A biplicit spectral‐collocation‐type ansatz for the numerical integration of partial differential equations with the transversal method of lines

Abstract

In the implicit formulation of the transversal method of lines, numerical instabilities (singular perturbation) do occur whenever "small" step sizes of the discretized variable have to be used for some reason. This problem can effectively be avoided if the derivatives with respect to the discretized variable are chosen using a combination of implicit and explicit methods (the biplicit method). This combination method uses piecewise defined trial functions involving a certain number of free parameters. The values of these parameters are found by the requirement that the trial functions approximate the solution of the implicit formulation of the method of lines. From a mathematical point of view, this spectral-collocation-type ansatz results in a multipoint boundaryvalue problem with added parameters to be determined. Two numerical examples are presented in order to illustrate the performance of the method. 0 1995 John Wiley & Sons, Inc.