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All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

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Abstract

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.

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Notes

  1. It is possible to derive the acoustic wave equation from (2.3). Indeed, if we differentiate with respect time the density equation and subtract it from the divergence of the momentum equation, we obtain

    $$\begin{aligned} \partial _{tt} \rho - \frac{\nabla ^2 p(\rho )}{\varepsilon ^2} = \nabla ^2:(\rho \mathbf u \otimes \mathbf u ), \end{aligned}$$

    and to \(\mathcal {O}(\varepsilon ^{0})\), we get (2.9).

  2. Note that this definition is different from the one used in Eq. (3.4) in 1D.

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Acknowledgements

The work has been partially supported by ITN-ETN Horizon 2020 Project ModCompShock, Modeling and Computation on Shocks and Interfaces, Project Reference 642768, by the National Group for Scientific Computing INdAM-GNCS Project 2017: Numerical methods for hyperbolic and kinetic equation and applications, and by the Department of Mathematics and Computer Science, University of Catania Piano triennale della Ricerca 2016–2018.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125, Catania, Italy

    S. Boscarino & G. Russo

  2. Heinrich-Heine-Universität Düsseldorf, Düsseldorf, North Rhine-Westphalia, Germany

    L. Scandurra

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  1. S. Boscarino
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  2. G. Russo
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  3. L. Scandurra
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Correspondence to G. Russo.

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S. Boscarino, G. Russo and L. Scandurra are members of the INdAM Research group GNCS.

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Boscarino, S., Russo, G. & Scandurra, L. All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics. J Sci Comput 77, 850–884 (2018). https://doi.org/10.1007/s10915-018-0731-9

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  • DOI: https://doi.org/10.1007/s10915-018-0731-9

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