Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

Abstract

In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut = (a(U )Ux)x. The basic idea is to add and subtract two equal terms a0Uxx on the right hand side of the partial differential equation, then to treat the term a0Uxx implicitly and the other terms (a(U )Ux)x − a0Uxx explicitly. We give stability analysis for the method on a simplified model by the aid of energy analysis, which gives a guidance for the choice of a0, i.e, a0 ≥ max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes. The optimal error estimate is also derived for the simplified model, and numerical experiments are given to demonstrate the stability, accuracy and performance of the schemes for nonlinear diffusion equations.

Accelerating exponential integrators to efficiently solve advection-diffusion-reaction equations

“…Note that in order to obtain an efficient scheme the choice of the parameter λ is crucial. In reference [29], in the context of IMEX schemes, the authors obtained satisfactory performances choosing λ as small as the stability condition permits. By following this approach, we also observed good results in the numerical validation performed in section 4.…”

“…A similar analysis can be performed also for other classes of schemes. For example, in reference [29] some IMEX schemes have been analyzed in a fully discretized context. In particular, for the well known backward-forward Euler method…”

“…The constantcoefficient approximation operator is treated implicitly (which can be done efficiently using, e.g., Fourier methods), while the nonlinear term is treated explicitly. Such schemes, known in the literature as explicit-implicit-null, were considered for instance in reference [29]. Clearly, the class of operators for which efficient linear solvers are known is not equivalent to the class of operators for which the matrix exponential (and related matrix functions) can be computed efficiently.…”

“…We investigate the choice of A and show that this can have a drastic influence on performance. We note that in reference [29], in the context of IMEX schemes, the authors employ the smallest amount of diffusion in A that is necessary to guarantee stability. We do observe a similar behavior for some test examples.…”

Stability of Spectral Collocation Schemes with Explicit-Implicit-Null Time-Marching for Convection-Diffusion and Convection-Dispersion Equations

“…Subsequently, it was used to stabilize the viscous free-surface dynamics of two liquid drops during coalescence [19], and the coarsening kinetics of interconnected two-phase mixtures [44]. In addition, the method has also been applied with success to, for example, the level set equations for mean curvature flow and motion by surface diffusion [38], the continuum models for the evolution of the molecular beam epitaxy growth [43], the Boltzmann kinetic equations and related problems with nonlinear stiff sources [22], the Cahn-Hilliard equations [37], the diffusion equations [42] with the local discontinuous Galerkin (LDG) method for spatial discretization, etc. These papers provide a clean description on the size of the constant a 0 , which is closely related to the equations discussed, the IMEX time-marching methods adopted and the auxiliary term added to and subtracted from the equation.…”