Kenneth H. Karlsen

Boris Andreianov, Kenneth H. Karlsen, Nils Henrik Risebro

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$Γ$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès \cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" $L^\infty$ weak-$\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz nonlinearities.

Simone Cifani, Espen R. Jakobsen, Kenneth H. Karlsen

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (Lévy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments.

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen et al.

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Giuseppe Maria Coclite, Kenneth H. Karlsen, Nils Henrik Risebro

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.

Fernando Betancourt, Raimund Bürger, Kenneth H. Karlsen

This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a non-linear function of the total mass to one side of the given position. This equation can be understood as a model of aggregation of the individuals of a population with the solution representing their local density. The aggregation mechanism is balanced by a degenerate diffusion term accounting for dispersal. In the strongly degenerate case, solutions of the non-local problem are usually discontinuous and need to be defined as weak solutions satisfying an entropy condition. A finite difference scheme for the non-local problem is formulated and its convergence to the unique entropy solution is proved. The scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Numerical examples illustrate the behaviour of entropy solutions of the non-local problem, in particular the aggregation phenomenon.

Helge Holden, Kenneth H. Karlsen, Trygve K. Karper

We analyze splitting algorithms for a class of two-dimensional fluid equations, which includes the incompressible Navier-Stokes equations and the surface quasi-geostrophic equation. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data are sufficiently regular.

Helge Holden, Kenneth H. Karlsen, Trygve K. Karper

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equations. In addition, Burgers-type equations with fractional diffusion as well as the KdV and Kawahara equations are covered. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data is sufficiently regular.

Kenneth H. Karlsen, Trygve K. Karper

We propose finite element methods for compressible barotropic Stokes systems. We state convergence results for these methods and outline their proofs. The principal tools of the proofs are higher integrability estimates for the discrete density, equations for the discrete effective viscous flux, and renormalized formulations of the numerical method for the density equation.

Thomas Christiansen, Kenneth H. Karlsen

We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the Itô formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-Itô correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.

Simone Cifani, Espen R. Jakobsen, Kenneth H. Karlsen

We propose, analyze, and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and whenever piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractal conservation laws.

Kenneth H. Karlsen, Trygve K. Karper

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div-curl form using the nonconforming Crouzeix-Raviart space. Our main result is that the finite element method converges to a weak solution. The main challenge is to demonstrate the strong convergence of the density approximations, which is mandatory in view of the nonlinear pressure function. The analysis makes use of a higher integrability estimate on the density approximations, an equation for the "effective viscous flux", and renormalized versions of the discontinuous Galerkin method.

Kenneth H. Karlsen, Trygve K. Karper

We propose and analyze a finite element method for a semi-stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nedelec spaces of the first kind. The continuity equation is approximated by a standard piecewise constant upwind discontinuous Galerkin scheme. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only div or curl conforming. (ii) To prove that the discontinuous Galerkin approximations converge strongly, which is required in view of the nonlinear pressure function. Tools involved in the analysis include a higher integrability estimate for the discontinuous Galerkin approximations, a discrete equation for the effective viscous flux, and various renormalized formulations of the discontinuous Galerkin scheme.