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Title: Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. theL2L^2norm of the numerical solution does not increase in time, under the time step conditionτ<#comment/>≤<#comment/>F(h/c,d/c2)\tau \le \mathcal {F}(h/c, d/c^2), with the convection coefficientcc, the diffusion coefficientdd, and the mesh sizehh. The functionF\mathcal {F}depends on the specific IMEX temporal method, the polynomial degreekkof the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomesτ<#comment/>≲<#comment/>h/c\tau \lesssim h/cin the convection-dominated regime and it becomesτ<#comment/>≲<#comment/>d/c2\tau \lesssim d/c^2in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

 
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Award ID(s):
2010107
PAR ID:
10498236
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
AMS
Date Published:
Journal Name:
Mathematics of Computation
Volume:
92
Issue:
344
ISSN:
0025-5718
Page Range / eLocation ID:
2475 to 2513
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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