Corrected operator splitting for nonlinear parabolic equations

We present a corrected operator splitting ( COS) method for solving nonline ar 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 fa ils to do so. COS is based on solving a conservation law for modeling conve ction, a heat-type equation for modeling diffusion and finally a certain re sidual conservation law for necessary correction. The residual equation rep resents 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 polyg onal method of Dafermos [J. Math. Anal. Appl., 38 ( 1972), pp. 33-41] const itutes an important part of our solution strategy. It is shown that COS gen erates a compact sequence of approximate solutions which converges to the s olution of the problem. Finally, some numerical examples are presented wher e we compare OS and COS methods with respect to accuracy.