this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their high-resolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semi-discrete formulation

During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multi-phase flow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic two-phase flow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convection-diffusion equation. Both theory and applications are discussed. The general background for the reservoir flow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct ...

Abstract. 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 conservationlaw for modeling convection, a heat type equation for modeling di usion, and nally a certain \residual" conservationlaw 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 counterbalancetheentropy 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. 0. 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 (1) ut + f(u)x = " (u)xx � u(x � 0) = u0(x) � x 2 R � t2 [0�T] � where u0(x), (u), and f(u) aregiven, su ciently smooth functions, and ">0 is a small scaling parameter.

. We first analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diffusion 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 finite difference scheme for the diffusion step. Numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic convection-diffusion equations. 0. Introduction. In this paper we consider viscous splitting methods for nonlinear, possibly strongly degenerate, convectiondiffusion equations of the form (1) ( @ t u + @ x f(u) = "@ 2 x A(u); (x; t) 2 QT = R \Theta h0; T i ; A 0 (u) 0; u(x; 0) = u 0 (x); where u(x; t) denotes the (scalar) unknown, u 0 (x) is a...

. Front tracking in combination with dimensional splitting is analyzed as a numerical method for scalar conservation laws in two space dimensions. An analytic error bound is derived, and convergence rates based on numerical experiments are presented. Numerical experiments indicate that large CFL numbers can be used without loss of accuracy for a wide range of problems. A new method for grid refinement is introduced. The method easily allows for dynamical changes in the grid, using, for instance, the total variation in each grid cell as a criterion for introducing new or removing existing refinements. Several numerical examples are included, highlighting the features of the numerical method. A comparison with a high-resolution method confirms that dimensional splitting with front tracking is a highly viable numerical method for practical computations. 1. Introduction Dimensional splitting, or fractional steps, has been widely used to extend one-dimensional numerical methods to multidim...

We consider a prototype two-phase fluid-flow model with capillary forces. The pressure equation is solved using standard finite-elements and multigrid techniques. The parabolic saturation equation is addressed via a novel corrected operator splitting approach. In typical applications, the importance of advection versus diffusion (capillary forces) may change rapidly during a simulation. The corrected splitting is designed so that any combination of advection and diffusion is resolved accurately. It gives a hyperbolic conservation law for modelling advection and a parabolic equation for modelling diffusion. The conservation law is solved by front tracking, which naturally leads to a dynamically defined residual flux term that can be included in the diffusion equation. The residual term ensures that self-sharpening fronts are given the correct structure. A Petrov--Galerkin finite-element method is used to solve the diffusion equation. We present several examples that demonstrate potenti...

We propose and analyse a numerical scheme for a class of advection dominated advection--diffusion--reaction equations. The scheme is essentially based on combining a front tracking method for conservation laws, which tracks shock curves defined by a varying velocity field, with a suitable operator splitting. The splitting is formulated for an equation in non-conservative form and consists of a nonlinear conservation law modelling advection, a heat equation modelling diffusion, and finally an ordinary differential equation modelling lower order processes. Since no CFL condition is associated with the front tracking scheme, our numerical scheme is unconditionally stable in the sense that the splitting time step is not restricted by the spatial discretization parameter. Nevertheless, it is observed that when the splitting time step is notably larger than the diffusion scale, the scheme can become too diffusive. This can be inferred with the fact that the entropy condition forces the hyp...

We present an accurate numerical method for a large class of scalar, strongly degenerate convection-diffusion equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation-consolidation processes. The method is based on splitting the convective and the diffusive terms. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion part is solved by an implicit--explicit finite difference scheme. In addition, one version of the implemented operator splitting method has a mechanism built in for detecting and correcting unphysical entropy loss, which may occur when the time step is large. This mechanism helps us gain a large time step ability for practical computations. A detailed convergence analysis of the operator splitting method was given in Part I. Here we present numerical experiments with the method for examples modelling secondary oil recovery and sedimentation-consolidation processes. We demonstrate that the splitting method resolves sharp gradients accurately, may use large time steps, has first order convergence, exhibits small grid orientation effects, has small mass balance errors, and is rather efficient.

. Many numerical methods for (one-dimensional) systems of convection-diffusion equations are based upon an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous s...

Operator splitting methods are often used to solve convection-diffusion problems of convection dominated nature. However, it is well known that such methods can produce significant (splitting) errors in regions containing self sharpening fronts. To amend this shortcoming, corrected operator splitting methods have been developed. These approaches use the wave structure from the convection step to identify the splitting error. This error is then compensated for in the diffusion step. The main purpose of the present work is to illustrate the importance of the correction step in the context of an inverse problem. The inverse problem will consist of estimating the fractional flow function in a one-dimensional saturation equation.