Corrected Operator Splitting for Nonlinear Parabolic Equations

Abstract

Abstract. We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of a convection-diffusion type. The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps, as opposed to a standard operator splitting (OS) which fails to do so. COS is based on solving a conservation law for modeling convection, a heat-type equation for modeling diffusion and finally a certain "residual" conservation law for necessary correction. The residual equation represents the entropy loss generated in the hyperbolic (convection) step. In OS the entropy loss manifests itself in the form of too wide shock fronts. The purpose of the correction step in COS is to counterbalance the entropy loss so that correct width of nonlinear shock fronts is ensured. The polygonal method of Dafermos [J. Math. Anal. Appl., 38 (1972), pp. 33-41] constitutes an important part of our solution strategy. It is shown that COS generates a compact sequence of approximate solutions which converges to the solution of the problem. Finally, some numerical examples are presented where we compare OS and COS methods with respect to accuracy.

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