The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) 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

Coronal heating distribution due to low-frequency wave-driven turbulence

The heating of the lower solar corona is examined using numerical simulations and theoretical models of magnetohydrodynamic turbulence in open magnetic regions. A turbulent energy cascade to small length scales perpendicular to the mean magnetic field can be sustained by driving with low-frequency Alfven waves reflected from mean density and magnetic field gradients. This mechanism deposits energy efficiently in the lower corona, and we show that the spatial distribution of the heating is determined by the mean density through the Alfven speed profile. This provides a robust heating mechanism that can explain observed high coronal temperatures and accounts for the significant heating (per unit volume) distribution below two solar radius needed in models of the origin of the solar wind. The obtained heating per unit mass on the other hand is much more extended indicating that the heating on a per particle basis persists throughout all the lower coronal region considered here.Comment: 19 pages, 5 figures. Accepted for publication in Ap

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Weak solutions for a gas liquid model relevant for describing gas-kick in oil wells

The article was originally published at; http://dx.doi.org/10.1137/100813932; it is made available here with permission.The purpose of this paper is to establish a local in time existence result for a compressible gas-liquid model. The model is a drift-flux model which is composed of two continuity equations and one mixture momentum equation supplemented with a slip relation in order to take into account the possibility of flows with unequal fluid velocities. The model is highly relevant for modeling of gas kick for oil wells, which in its worst case can lead to blowout scenarios. The mathematical study of such kinds of models is important for the development of simulation tools that can be employed for increased control of deep-water well operations. The liquid phase is assumed to be incompressible whereas the gas is described by a polytropic equation of state. The model is studied in a framework previously used for investigations of the single-phase compressible Navier–Stokes model. New challenges arise due to the appearance of a generalized pressure term that depends on fluid masses as well as gas velocity. The local existence result is obtained by introducing a suitable transformation along the line of the works [S. Evje and K. H. Karlsen, Commun. Pure Appl. Anal., 8 (2009), pp. 1867–1894, S. Evje, T. Flåtten, and H. A. Friis, Nonlinear Anal., 70 (2009), pp. 3864–3886] in a free boundary setting. This allows us to obtain sufficient pointwise control of the gas and liquid masses. The estimates are rather delicate as they must be fine enough to control a possible singular behavior associated with the pressure law as well as the slip relation. The existence result is obtained under the assumption of a sufficient small time interval combined with suitable assumptions on the regularity of the initial data, the parameters that control, respectively, the behavior of the initial masses at the boundaries of the flow domain and the decay properties of the viscosity term

On Strongly Degenerate Convection-Diffusion Problems Modeling Sedimentation-Consolidation Processes

. We investigate initial-boundary value problems for a quasilinear strongly degenerate convection-diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentation-consolidation processes. Existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second order equations of Kruzkov's method of "doubling of the variables". The application to a sedimentation-consolidation model is illustrated by two numerical examples. 1. Introduction In this paper, we consider quasilinear strongly degenerate parabolic equations of the type @ t u + @ x (q(t)u + f(u)) = @ 2 x A(u); (x; t) 2 Q T ; A(u) := Z u 0 a(s) ds; a(u) 0; (1.1) where QT :=\Omega \Theta T,\Omega := (0;..