A Note on Viscous Splitting of Degenerate Convection-diffusion Equations

We establish L convergence of a viscous splitting method for nonlinear possibly strongly degenerate convection-di usion problems. Since we allow the equations to be strongly degenerate, solutions can be discontinuous and they are not, in general, uniquely determined by their data. We thus consider entropy weak solutions realized by the vanishing viscosity method. This notion is broad enough to also include non-degenerate parabolic equations as well as hyperbolic conservation laws. It thus provides a suitable \L type" framework for analyzing numerical schemes for convection-di usion problems that are designed to handle various balances of convective and di usive forces. We present a numerical example which shows that our splitting scheme has such \design". 0. Introduction It is well known that accurate modeling of convective and di usive processes is one of the most ubiquitous and challenging tasks in the numerical approximation of partial di erential equations. This is partly because of the problems themselves, their widespread occurrence, as well as their close association with hyperbolic conservation laws. Nonlinear convection-di usion equations arise in a variety of applications, ranging from models of turbulence [10], via tra c ow [48] and nancial modeling [8], to two phase ow in porous media [11]. A convection-di usion 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 [62] or the Navier-Stokes equations. Such equations also appear in, to mention a few, polymer chemistry [12], combustion modeling [51], modeling of semi-conductor devices [50], and in models for transport of solutes in ground water and surface water [6]. One objective of this paper is to establish convergence of a viscous splitting method for nonlinear possibly strongly degenerate convection-di usion problems. First, convergence of the splitting will be established for the following initial value problem (1) @tu+ @xf(u) = "@x [a(u)@xu] ; u(x; 0) = u0(x); (x; t) 2 QT ; where u : QT R h0; T ]! Rdenotes the unknown; u0 : R! [m;M ], f : [m;M ]! R, and a : [m;M ]! [0;1i are given smooth, bounded functions; and " > 0 is a scaling parameter. Secondly, the initial-boundary value problem will be discussed and we show how to extend our method of proof to cover also this problem. When (1) is non-degenerate, i.e., a(u) > 0 8u, the mathematical theory is well known [54]. In particular, the initial value problem then admits a unique classical solution. This contrasts with the case where (1) is allowed to degenerate at certain points. The solution is then not necessarily smooth [1,2,3,53] and weak solutions must be sought. Almost all previous works on mathematical analysis focused mainly on the special case where the equation possesses only one or two point degeneracy and often only non-negative continuous solutions were considered [21,29,45,46]. The simplest examples are perhaps provided by the porous medium equation; @tu = "@ 2 x (u ), m > 1, and the convective porous medium equation; @tu + @x (u ) = "@ x (u ), n;m > 1. To the best of our knowledge, the most general result on the existence and uniquness of weak solutions for (1) in the class of bounded and measurable functions is provided by Zhao [42]. The essential condition for uniqueness is that the function (2) A(u) = Z u 0 a( ) d