Corrected Operator Splitting for Nonlinear Parabolic Equations

We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of convection-diiusion 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 diiusion, 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. In this paper we introduce a novel operator splitting method for constructing approximate solutions to nonlinear parabolic convection-diiusion problems of the form (1) u t + f(u) x = ""(u) xx ; u(x; 0) = u 0 (x); x 2 R; t 2 0; T]; where u 0 (x), (u), and f(u) are given, suuciently smooth functions, and " > 0 is a small scaling parameter. Partial diierential equations from mathematical physics sometimes appear in the non-conservative form (2) u t + f(u) x = " (d(u)u x) x ; where we can assume that d(u) is a strictly positive function, so that (2) is parabolic and admits classical solutions. The mixed hyperbolic/parabolic case (d(u) 0) is addressed in 12]. In the parabolic context we can obviously write (2) in conservative form (1), so that any solution strategy presented for (1) applies equally well to (2). Consequently, we choose to work with (1) in this paper. Existence and uniqueness of a classical solution to (1) is well known, see for example 29,30]. Furthermore, the notion of a classical solution coincides with the notion of a weak solution for parabolic equations such as (1), see 29]. Equations such as (1) arise in a variety of applications, ranging from models of turbulence 4], via traac ow 28] and nancial modeling 3], to two phase ow …