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A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model

Juntao Huang

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, P. R. China

E-mail Address: huangjt13@mails.tsinghua.edu.cn

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and

Chi-Wang Shu

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

E-mail Address: shu@dam.brown.edu

Corresponding author

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https://doi.org/10.1142/S0218202517500099Cited by:22 (Source: Crossref)

Abstract

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal.42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

Communicated by F. Brezzi

Keywords:

AMSC: 65M60, 65M12

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