Diffusive BGK approximations for nonlinear multidimensional parabolic equations

Abstract

We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diffusive conservation laws. We prove the well-posedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution recently obtained by Carrillo. Examples of such BGK models are provided.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…This property follows by a straightforward comparison of the formulas (21) and (22) for the discrete gradient with the reconstruction formulas of the next lemma.…”

“…Under this convention, Lemma 3.1 below holds true, so that formulas (22), (23)- (25) below still yield consistent discrete gradient and discrete divergence operators which enjoy the discrete duality property [7]. But the discrete entropy dissipation inequalities of Proposition 4.2 would fail, which undermines the subsequent convergence analysis.…”

“…For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]). …”

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

“…The (very recent) literature include papers by Evje and Karlsen [18], Holden et al [25], and Holden et al [26] on operator splitting methods (see also the lecture notes by Espedal and Karlsen [15]); Evje and Karlsen [16,17,19,20] on upwind difference schemes; Kurganov and Tadmor [35] on central difference schemes; Bouchut et al [2] on kinetic BGK schemes; Afif and Amaziane [1] and Ohlberger [39] on finite volume methods; and Cockburn and Shu [11] on the local discontinuous Galerkin method. Strictly speaking, the authors of [1,11,35] do not analyze their numerical methods within an entropy solution framework.…”

On a relaxation approximation of the incompressible Navier-Stokes equations

“…For other diffusive kinetic models and approximations, we refer to [15], [13], [12]. A general class of kinetic approximations for (possibly degenerate) parabolic equations in multi-D has been considered in [4], [1]. Let us also point out that the same scaling was used in [16] to analyze the time-asymptotic limit of the Jin and Xin relaxation model [14], towards the fundamental solution of the diffusive Burgers equation.…”