## On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations

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Jingwei Hu and Ruiwen Shu
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## Abstract:

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: jingweihu@purdue.edu
**Ruiwen Shu**- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 1204137
- Email: rshu@cscamm.umd.edu
- 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: https://doi.org/10.1090/mcom/3602
- MathSciNet review: 4194158