Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Abstract

Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h 1/4 | ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator. (2000): 35L65, 35Q35, 65M15 Mathematics Subject Classification

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

“…To finish with the bibliography, let us also refer the reader to [20,21,22,28,32,55] for the related topic of error estimates for numerical approximations.…”

On numerical methods and error estimates for degenerate fractional convection–diffusion equations

“…When it comes to nonlocal convection-diffusion equations, the literature is very recent and not yet very extensive. The paper [15] introduce finite volume schemes for radiation hydrodynamics equations, a model where L µ is a nonlocal derivative of order 0. Then fractional conservation laws are discretized in [17,13,12] with finite difference, discontinuous Galerkin, and spectral vanishing viscosity methods respectively.…”

“…Then fractional conservation laws are discretized in [17,13,12] with finite difference, discontinuous Galerkin, and spectral vanishing viscosity methods respectively. In [15,13] Kuznetsov type error estimates are given, but only for integrable Lévy measures or measures like (1.3) with λ < 1. Both of these results can be obtained through the framework of this paper.…”

The nonrelativistic limit in radiation hydrodynamics: I. Weak entropy solutions for a model problem

“…See [9,12] for the full system of equations. In [2,3], the following model It was pointed out in [2,3] that the relation of system (1.1) to the full system of radiation hydrodynamics is similar to the relation of scalar nonlinear conservation laws to the Euler equations of compressible hydrodynamics. In particular, u is a lumped variable for the original hydromechanical unknowns (density, velocity, and temperature) and I is the radiation intensity.…”