A note on the stability of implicit-explicit flux-splittings for stiff systems of hyperbolic conservation laws

We analyze the stability of implicit-explicit flux-splitting schemes for stiff systems of conservation laws. In particular, we study the modified equation of the corresponding linearized systems. We first prove that symmetric splittings are stable, uniformly in the singular parameter ε. Then we study non-symmetric splittings. We prove that for the barotropic Euler equations, the Degond–Tang splitting [Degond & Tang, Comm. Comp. Phys. 10 (2011), pp. 1–31] and the Haack– Jin–Liu splitting [Haack, Jin & Liu, Comm. Comp. Phys. 12 (2012), pp. 955 980], and for the shallow water equations the recent RS-IMEX splitting are strictly stable in the sense of Majda–Pego. For the full Euler equations, we find a small instability region for a flux splitting introduced by Klein [Klein, J. Comp. Phys. 121 (1995), pp. 213–237], if this splitting is combined with an IMEX scheme as in [Noelle, Bispen, Arun, Lukác̆ová, Munz, SIAM J. Sci. Comp. 36 (2014), pp. B989–B1024].

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