Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems

Wang, Haijin; Shu, Chi‐Wang; Zhang, Qiang

doi:10.1137/140956750

Cited by 123 publications

(80 citation statements)

References 22 publications

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“…In this paper we extend our previous work [14] and continue to analyze the stability and error estimates of some fully discrete local discontinuous Galerkin (LDG) schemes coupled with implicit-explicit (IMEX) time discretization, for solving a semilinear convection-diffusion problem with periodic boundary condition in one dimension.…”

Section: Introductionmentioning

confidence: 83%

“…For the readers who are interested in the proof, we refer to [19]. Lemma 2.3 given in [14] builds up an important relationship between the auxiliary variable and the prime variable, which plays a key role in the stability analysis. …”

Section: Properties Of the Ldg Spatial Discretizationmentioning

confidence: 99%

“…In [14] we have discussed the LDG scheme coupled with three specific Runge-Kutta type IMEX schemes given in [1,3], for solving linear advection-diffusion problems. We have shown that those schemes are unconditionally stable in the sense that the time step s is only required to be upper bounded by a fixed constant which depends solely on the coefficients of advection and diffusion, but is independent of the mesh size.…”

Section: Introductionmentioning

confidence: 99%

“…As we pointed out in [14], there is one more stage for the explicit part than the implicit part in the third order IMEX RK scheme in [3], which makes the error analysis more complicated than the linear case [14], where we only considered the upwind numerical flux for the convection part, so that we can find a proper projection (Gauss-Radau projection) to eliminate the cell interface errors to obtain optimal error accuracy. While for the nonlinear problems, if we consider a general monotone numerical flux, it is difficult to find such a projection to eliminate the cell interface errors, so we would like to resort to the technique used in [20,21] to carry out the error analysis for the third order IMEX LDG scheme.…”

Section: Introductionmentioning

confidence: 99%

“…In the analysis, an a priori assumption is required, and the idea in [12] to divide the total error into spatial error and temporal error is also adopted. The error estimates for the first and second order IMEX LDG schemes presented in [14] will not be presented in this paper to save space, because the numbers of explicit stages and implicit stages for these two lower order schemes are the same, hence their analysis is simpler than the third order scheme; see Remark 3.…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

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2016

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a b s t r a c tThe main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step s is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size h. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

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Section: Introductionmentioning

confidence: 83%

Section: Properties Of the Ldg Spatial Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discontinuous Galerkin Methods for Computational Fluid Dynamics

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2017

Encyclopedia of Computational Mechanics Second Edition

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The discontinuous Galerkin (DG) methods are locally conservative, high‐order accurate, robust methods that can easily handle elements of arbitrary shapes, irregular meshes with hanging nodes, and polynomial approximations of different degrees in different elements. These properties, which render them ideal for hp ‐adaptivity in domains of complex geometry, have brought them to the main stream of computational fluid dynamics. We study the properties of the DG methods as applied to a wide variety of problems arising in fluid dynamics with a special emphasis on linear, symmetric positive hyperbolic systems, the Euler equations of gas dynamics, purely elliptic problems, and the incompressible and compressible Navier–Stokes equations. In each instance, we discuss the main properties of the methods, display the mechanisms that make them work so well, and present numerical experiments showing their performance.

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Discontinuous Galerkin Time‐Domain Method in Electromagnetics: From Nanostructure Simulations to Multiphysics Implementations

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In the last few decades, the discontinuous Galerkin time-domain (DGTD) method has become widely popular in various fields of engineering due the fact that it benefits from computational advantages that come with finite volume and finite element formulations. Similarly, in the field of computational electromagnetics, the superiority of the DGTD method has been quickly recognized after first few works on its formulation and 1 implementation to solve Maxwell equations. With further developments in more recent years, the DGTD method has become one of the preeminent solutions to tackle a wide variety of challenging large scale electromagnetic problems including those that require multiphysics modeling.This chapter starts with a brief introduction to the DGTD method. This introduction provides the fundamentals of numerical flux, discretization techniques that rely on vector and nodal basis functions, and incorporation of absorbing boundary conditions. This is followed by descriptions of a time-domain boundary integral(TDBI) scheme, which replaces absorbing boundary conditions within the DGTD method, and a multi-step time integration technique, which uses different time step sizes for the DGTD and TDBI parts. Numerical results show that both techniques significantly improve the efficiency, accuracy, and stability of the traditional DGTD method. Then, the chapter continues with the applications of the DGTD method to several real-life practical problems. More specifically, it describes various novel techniques developed to enable the application of the DGTD method to electromagnetic analysis of nanostructures and graphene-based devices, and multiphysics simulation of optoelectronic antennas and source generators. For each application, several numerical examples are provided to demonstrate the accuracy, efficiency, and robustness of the developed techniques.

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Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems

Cited by 123 publications

References 22 publications

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

Discontinuous Galerkin Methods for Computational Fluid Dynamics

Discontinuous Galerkin Time‐Domain Method in Electromagnetics: From Nanostructure Simulations to Multiphysics Implementations

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