Corrected Operator Splitting

We present a corrected operator splitting COS method for solving nonlinear parabolic equations of convection di usion type The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps as opposed to 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 di usion and nally 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 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 Introduction In this paper we introduce a novel operator splitting method for constructing approximate solutions to nonlinear parabolic convection di usion problems of the form ut f u x u xx u x u x x R t T where u x u and f u are given su ciently smooth functions and is a small scaling parameter Partial di erential equations from mathematical physics sometimes appear in the non conservative form ut f u x d u ux x where we can assume that d u is a strictly positive function so that is parabolic and admits classical solutions The mixed hyperbolic parabolic case d u is addressed in In the parabolic context we can obviously write in conservative form so that any solution strategy presented for applies equally well to Consequently we choose to work with in this paper Existence and uniqueness of a classical solution to is well known see for example Furthermore the notion of a classical solution coincides with the notion of a weak solution for parabolic equations such as see Equations such as arise in a variety of applications ranging from models of turbulence via tra c ow and nancial modeling to two phase ow in porous media Equation can also be viewed as a model problem for a system of convection di usion equations such as three phase ow in porous media or the Navier Stokes equations Of particular importance is the case where convection dominates di usion i e is small compared with other scales in This is often the case in models of two phase ow in oil reservoirs Accurate numerical simulations of such models are consequently often complicated by both unphysical oscillations and numerical di usion If then is almost hyperbolic and it is natural to exploit this when constructing numerical methods A widely used strategy is viscous operator splitting OS henceforth that is splitting into a hyperbolic conservation law and a parabolic heat equation each of which is solved by some proper numerical scheme This approach or at least certain variations on this approach has indeed been taken by several authors we mention Beale and Majda Douglas and Russell Espedal and Ewing Dahle Dawson Karlsen and Risebro and more recently Evje and Karlsen In a characteristic element method is used to solve the hyperbolic part of In error estimates are obtained for a linear version of In it is shown that the viscous splitting method converges to the solution of also in multi dimensions in the case of linear di usion a Lipschitz continuous ux function and any discontinuous initial function of bounded variation Convergence results for the viscous splitting method for one dimensional parabolic equations with a nonlinear possibly strongly degenerate di usion term are obtained in Mathematics Subject Classi cation L