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 ...

We study nonlinear degenerate parabolic equations where the flux function f(x, t, u) does not depend Lipschitz continuously on the spatial location x. By properly adapting the “doubling of variables” device due to Kruˇzkov [24] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form k(x)f(u), where k(x) is a vector-valued function and f(u) is a scalar function.

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 describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convection-diffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.

. 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...

We apply the method of operator splitting on the generalized Korteweg-de Vries (KdV) equation u t +f(u)x +"uxxx = 0, by solving the nonlinear conservation law u t +f(u)x = 0 and the linear dispersive equation u t + "uxxx = 0 sequentially. We prove that if the approximation obtained by operator splitting converges, then the limit function is a weak solution of the generalized KdV equation. Convergence properties are analyzed numerically by studying the eect of combining different numerical methods for each of the simplified problems.

convergence of a viscous splitting method for nonlinear possibly strongly degenerate convection-diffusion 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 1 type " framework for analyzing numerical schemes for convection-diffusion problems that are designed to handle various balances of convective and diffusive forces. We present a numerical example which shows that our splitting scheme has such "design". It is well known that accurate modeling of convective and diffusive processes is one of the most ubiquitous and challenging tasks in the numerical approximation of partial differential equations. This is partly because of the problems themselves, their widespread occurrence, as well as their close association with hyperbolic

Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.

There is increasing interest in the analysis of biological tissue, its organization and its dynamics with the help of mathematical models. In the ideal case emergent properties on the tissue scale can be derived from the cellular scale. However, this has been achieved in rare examples only, in particular, when involving high-speed migration of cells. One major difficulty is the lack of a suitable multiscale simulation platform, which embeds reaction-diffusion of soluble substances, fast cell migration and mechanics, and, being of great importance in several tissue types, cell flow homeostasis. In this paper a step into this direction is presented by developing an agent-based mathematical model specifically designed to incorporate these features with special emphasis on high speed cell migration. Cells are represented as elastic spheres migrating on a substrate in lattice-free space. Their movement is regulated and guided by chemoattractants that can be derived from the substrate. The diffusion of chemoattractants is considered to be slower than cell migration and, thus, to be far from equilibrium. Tissue homeostasis is not achieved by the balance of growth and death but by a flow equilibrium of cells migrating in and out of the tissue under consideration. In this sense the number and the distribution of the cells in the tissue is a result of the model and not part of the assumptions. For purpose of demonstration of the model properties and functioning, the model is applied to a prominent example of tissue in a cellular flow equilibrium, the secondary lymphoid tissue. The experimental data on cell speed distributions in these tissues can be reproduced using reasonable mechanical parameters for the simulated cell migration in dense tissue.

. A new front tracking method is developed for the variable coefficient equation u t + V (x; t)f(u) x = 0. The method is a generalization of Dafermos' method for the constant coefficient case and is well-defined also for certain discontinuous velocities V . We give an explicit inequality stating the stability with respect to flux function, velocity, and initial data. The numerical method is unconditionally stable and has linear convergence. It is well suited for numerical calculations, as is demonstrated in three examples. 1. Introduction We consider the nonlinear, variable coefficient equation u t + V (x; t)f(u) x = 0; u(x; 0) = u 0 (x): (1) This equation describes the nonlinear advection of a scalar quantity u in a one-dimensional velocity field V (x; t). A motivation for this study is that (1) occurs as an essential step in various operator splittings; for instance, dimensional splitting for multi-dimensional, variable coefficient equations [11], two-phase flow with capillary for...