Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations

We rst analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diiusion equations. Due to strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. Hence weak solutions satisfying an entropy condition are sought. We then propose and analyse a fully discrete splitting method which employs a front tracking method for the convection step and a nite diierence scheme for the diiusion step. Numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic convection-diusion equations. In this paper we consider viscous splitting methods for nonlinear, possibly strongly degenerate, convection-diiusion equations of the form (1) (@ t u + @ x f(u) = "@ 2 x A(u); (x; t) 2 Q T = R h0; Ti; A 0 (u) 0; u(x; 0) = u 0 (x); where u(x; t) denotes the (scalar) unknown, u 0 (x) is a given function of bounded variation, f(u) and A(u) are given locally smooth bounded functions, and " > 0 is a (small) scaling parameter. Convection-diiusion equations arise in a variety of applications, among others turbulence, traac ow, nan-cial modelling and front propagation. Such equations also constitute an important part of a system of equations describing two phase ow in oil reservoirs 6] as well as a system of equations describing sedimentation processes used for solid-liquid separation in industrial applications 5,4]. When (1) is convection dominated, which is often the case in reservoir simulation, it is well known that conventional numerical methods exhibit non-physical oscillations and/or excessive numerical diiusion in the vicinity of moving shock fronts. An underlying design principle for many successful numerical methods for equations such as (1), is viscous operator splitting. That is, one splits the time evolution into two partial steps in order to separate the eeects of convection and diiusion (viscosity). Variations on the viscous splitting approach have indeed been taken in analysed a fully discrete splitting method for (1) in the linear diiusion case A(u) u. Their discrete method was based on a front tracking scheme for the convection step and a nite element scheme for the diiusion step. A variant of this method has also been implemented in a prototype two-dimensional black oil reservoir simulator 19]. The idea is to treat multi-dimensional problems by means of dimensional splitting. The resulting method allowed for very long time steps due to the …