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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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On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations
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by Jingwei Hu and Ruiwen Shu HTML | PDF
Math. Comp. 90 (2021), 641-670 Request permission


Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon =O(1)$) and limiting regime ($\varepsilon \rightarrow 0$). However, in the intermediate regime (say, $\varepsilon =O(\Delta t)$), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptotic or stability analysis). In this work, we prove the uniform accuracy – an optimal a priori error bound – of a class of IMEX multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for linear hyperbolic systems with stiff relaxation. The proof is based on the energy estimate with a new multiplier technique. For nonlinear hyperbolic and kinetic equations, we numerically verify the same property using a series of examples.
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Additional Information
  • Jingwei Hu
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • MR Author ID: 933436
  • Email:
  • Ruiwen Shu
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 1204137
  • Email:
  • Received by editor(s): January 24, 2020
  • Received by editor(s) in revised form: June 24, 2020, August 13, 2020, and August 17, 2020
  • Published electronically: November 24, 2020
  • Additional Notes: The first author’s research was supported in part by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152. The second author’s research was supported in part by NSF grants DMS-1613911, RNMS-1107444 (KI-Net) and ONR grant N00014-1812465.
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 641-670
  • MSC (2020): Primary 35L03, 82C40, 65L04, 65L06, 65M12
  • DOI:
  • MathSciNet review: 4194158