Papers by Kenneth Karlsen
The IMA Volumes in Mathematics and its Applications, 2000
Hyperbolic Problems: Theory, Numerics, Applications, 2001
We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starti... more We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This 2 × 2 system of conserva- tion laws is coupled to the integral equation through natural coupling con- ditions accounting for the flow
Mathematical Finance, 2001
Summary. In this paper we present and analyse certain discrete approxima- tions of solutions to s... more Summary. In this paper we present and analyse certain discrete approxima- tions of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form ∂tu + ∂xf (u )= ∂xA(b(u)∂xu) ,u (x, 0) = u0(x),
Hyperbolic Problems: Theory, Numerics, Applications, 1999
Journal of Multivariate Analysis, 2000
We consider the initial value problem for degenerate viscous and inviscid scalar conserva- tion l... more We consider the initial value problem for degenerate viscous and inviscid scalar conserva- tion laws where the flux function depends on the spatial location through a \rough" coecient function k(x). We show that the Engquist-Osher (and hence all monotone) nite dierence approximations con- verge to the unique entropy solution of the governing equation if, among other demands, k0 is in
Summary. We first analyse a semi-discrete operator splitting method for nonlinear, possibly stron... more Summary. 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 en- tropy condition are sought. We then propose and analyse a fully discrete split- ting method which employs a front tracking
For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions ... more For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic-parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical
We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations ... more 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 abil- ity 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
We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-d... more We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and, in general, are not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition,
We study nonlinear degenerate parabolic equations where the flux function f(x,t,u) does not depen... more 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 Kruzkov (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
Hyperbolic Problems: Theory, Numerics, Applications, 1999
. A front tracking method is used to construct weak solutions toscalar conservation laws with two... more . A front tracking method is used to construct weak solutions toscalar conservation laws with two kinds of boundary conditions --- Dirichletconditions and a novel zero flux (or no-flow) condition. The construction leadsto an efficient numerical method. The main feature of the scheme is that thereis no stability condition correlating the spatial and temporal discretizationparameters. The analysis uses the traditional method of proving compactnessvia Helly's theorem as well as the more...
Authors: Prof. Kenneth H. Karlsen Prof. Nils Henrik Risebro Centre of Mathematics for Application... more Authors: Prof. Kenneth H. Karlsen Prof. Nils Henrik Risebro Centre of Mathematics for Applications Department of Mathematics University of Oslo PO Box 1053 Blindern N0-0316 0SL0 NORWAY Prof. Helge Holden Department of Mathematical Sciences Norwegian University of ...
Networks and Heterogeneous Media, 2006
ABSTRACT We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. T... more ABSTRACT We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This 2 × 2 system of conserva- tion laws is coupled to the integral equation through natural coupling con- ditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incom- pressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reser- voir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.
Networks and Heterogeneous Media, 2008
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Papers by Kenneth Karlsen