Showing papers by "Nils Henrik Risebro published in 2000"

Corrected Operator Splitting for Nonlinear Parabolic Equations

Abstract: We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of a convection-diffusion type. The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps, as opposed to a 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 diffusion and finally 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 [ J. Math. Anal. Appl., 38 (1972), pp. 33--41] 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.

A fast marching method for reservoir simulation

Abstract: We present a fast marching level set method for reservoir simulation based on a fractional flow formulation of two-phase, incompressible, immiscible flow in two or three space dimensions The method uses a fast marching approach and is therefore considerably faster than conventional finite difference methods The fast marching approach compares favorably with a front tracking method as regards both efficiency and accuracy In addition, it maintains the advantage of being able to handle changing topologies of the front structure

Front Tracking and Operator Splitting for Nonlinear Degenerate Convection-Diffusion Equations

Abstract: 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.