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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation
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by Haijin Wang, Fengyan Li, Chi-Wang Shu and Qiang Zhang
Math. Comp. 92 (2023), 2475-2513
Published electronically: May 15, 2023


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. the $L^2$ norm of the numerical solution does not increase in time, under the time step condition $\tau \le \mathcal {F}(h/c, d/c^2)$, with the convection coefficient $c$, the diffusion coefficient $d$, and the mesh size $h$. The function $\mathcal {F}$ depends on the specific IMEX temporal method, the polynomial degree $k$ of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes $\tau \lesssim h/c$ in the convection-dominated regime and it becomes $\tau \lesssim d/c^2$ in 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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Bibliographic Information
  • Haijin Wang
  • Affiliation: School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu Province, People’s Republic of China
  • MR Author ID: 1022956
  • ORCID: 0000-0001-7912-4548
  • Email:
  • Fengyan Li
  • Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180
  • MR Author ID: 718718
  • Email:
  • Chi-Wang Shu
  • Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 242268
  • Email: chi-wang$_$
  • Qiang Zhang
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, People’s Republic of China
  • MR Author ID: 637183
  • ORCID: 0000-0003-1796-7042
  • Email:
  • Received by editor(s): June 4, 2022
  • Received by editor(s) in revised form: January 2, 2023, and February 23, 2023
  • Published electronically: May 15, 2023
  • Additional Notes: The first author was supported by NSFC grants 12071214 and 11871428, and Natural Science Research Program for Colleges and Universities of Jiangsu Province grant 20KJB110011. The second author was supported by NSF grant DMS-1913072. The third author was supported by NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055. The fourth author is the corresponding author, who was supported by NSFC grant 12071214.
  • © Copyright 2023 American Mathematical Society
  • Journal: Math. Comp. 92 (2023), 2475-2513
  • MSC (2020): Primary 65M12, 65M60
  • DOI: