Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Abstract

Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

Linearly Implicit Approximations of Diffusive Relaxation Systems

“…In [13] it was also shown that the time semi-discrete scheme can be seen as a generalization of a class of numerical schemes based on the so called nonlinear Chernoff formula (see [5]). Recently, in [12], explicit schemes based on Runge-Kutta Discontinuous Galerkin were also developed and analysed.…”

“…The previous Theorem asserts a stability result only for p(u h ), and not for u h ; we need to add the further hypothesis of strict monotonicity of the nonlinearity p in (x 0 , +∞) for some x 0 to grant the stability also of u h (see [32] and [12] for further details).…”

“…Usually, the numerical schemes developed for the relaxation systems are explicit, as in [2,[12][13][14]28] and this can be computationally expensive in the case of diffusive relaxation systems due to the "natural" parabolic stability restriction t ≤ ch 2 (1) that must be imposed on the time step t , where h is the length of the grid in space.…”

A structure-preserving discontinuous Galerkin scheme for the Fisher–KPP equation

“…Positivitypreserving DG schemes for parabolic equations were developed in, e.g., [9,15,23,33,36]. The positivity preservation is ensured by using a special slope limiter (as in [9,15]), together with a strong stability preserving Runge-Kutta time discretization (as in [33,36]), while in [23], the positivity of the discrete solution is enforced through a reconstruction algorithm, based on positive cell averages. As far as we know, the use of an exponential transformation to ensure the positivity of the discrete solutions within a DG scheme is new.…”

A structure-preserving discontinuous Galerkin scheme for the Fischer-KPP equation

“…The design of structure-preserving DG methods is a rather recent topic. Positivitypreserving DG schemes for parabolic equations were developed in, e.g., [9,15,23,33,36]. The positivity preservation is ensured by using a special slope limiter (as in [9,15]), together with a strong stability preserving Runge-Kutta time discretization (as in [33,36]), while in [23], the positivity of the discrete solution is enforced through a reconstruction algorithm, based on positive cell averages.…”