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A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics

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Abstract

In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where the Euler equations have been solved solved using a semi-implicit finite volume scheme based on the flux-vector splitting method recently proposed by Toro and Vázquez-Cendón (Comput Fluids 70:1–12, 2012). In our scheme, the nonlinear convective terms are discretized explicitly, while the pressure terms are discretized implicitly. As a consequence, the time step is restricted only by a mild CFL condition based on the fluid velocity, which makes this method particularly suitable for simulations in the low Mach number regime. However, the conservative formulation of the scheme, together with the novel subcell finite volume limiter allows also the numerical simulation of high Mach number flows with shock waves. Inserting the discrete momentum equation into the discrete total energy conservation law yields a mildly nonlinear system with the scalar pressure as the only unknown. Due to the use of staggered meshes, the resulting pressure system has the most compact stencil possible and can be efficiently solved with modern iterative methods. In order to deal with shock waves or steep gradients, the new semi-implicit DG scheme proposed in this paper includes an a posteriori subcell finite volume limiting technique. This strategy was first proposed by Dumbser et al. (J Comput Phys 278:47–75, 2014) for explicit DG schemes on collocated grids and is based on the a posteriori MOOD algorithm of Clain, Loubère and Diot. Recently, this methodology was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in Ioriatti and Dumbser (Appl Numer Math 135:443–480, 2019). Within the MOOD approach, an unlimited DG scheme first produces a so-called candidate solution for the next time level \(t^{n+1}\). Later on, the control volumes with a non-admissible candidate solution are identified by using physical and numerical detection criteria, such as the positivity of the solution, the absence of floating point errors and the satisfaction of a relaxed discrete maximum principle (DMP). Then, in the detected troubled cells a more robust first order semi-implicit finite volume (FV) method is applied on a sub-grid composed of \(2P + 1\) subcells, where P denotes the polynomial degree used in the DG scheme. For that purpose, the nonlinear convective terms are recomputed in the troubled cells using an explicit finite volume scheme on the subcell level. Also the linear system for the pressure needs to be assembled and solved again, but where now a low order semi-implicit finite volume scheme is used on the sub-cell level in all troubled DG elements, instead of the original high order DG method. Finally, the higher order DG polynomials are reconstructed from the piecewise constant subcell finite volume averages and the scheme proceeds to the next time step. In this paper we present, discuss and test this novel family of methods and simulate a set of classical numerical benchmark problems of compressible gasdynamics. Great attention is dedicated to 1D and 2D Riemann problems and we also show that for these test cases the scheme responds appropriately in the presence of shock waves and does not produce non-physical spurious numerical oscillations.

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Notes

  1. When the index i is omitted in the vector of degrees of freedom we intend the entire set of all degrees of freedom of all elements.

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Acknowledgements

The authors acknowledge funding from the Istituto Nazionale di Alta Matematica (INdAM) via the GNCS group and the program Young Researchers Funding 2018 via the research project Semi-implicit structure preserving schemes for continuum mechanics. MD acknowledges the financial support received from the Italian Ministry of Education, University and Research (MIUR) in the frame of the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (Grant L. 232/2016) and in the frame of the PRIN 2017 Project Innovative numerical methods for evolutionary partial differential equations and applications. MD has also received funding from the University of Trento via the Strategic Initiative Modeling and Simulation.

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  1. Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123, Trento, Italy

    Matteo Ioriatti & Michael Dumbser

  2. Institut de Mathèmatiques de Bordeaux (IMB), 33405, Talence, France

    Raphaël Loubère

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  1. Matteo Ioriatti
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  2. Michael Dumbser
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  3. Raphaël Loubère
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Appendix A: Matrices and Tensors for the 1D Semi-implicit Schemes

Appendix A: Matrices and Tensors for the 1D Semi-implicit Schemes

1.1 Appendix A.1: Matrices and Tensors for the 1D Semi-implicit DG Method

$$\begin{aligned} \mathbf {M}= & {} \int \limits _0^1{\varvec{\varphi }}(\xi )\varvec{\varphi }(\xi )d\xi \end{aligned}$$
(A.1)
$$\begin{aligned} \mathbf {K}= & {} \int \limits _0^1{\varvec{\varphi }}'(\xi )\varvec{\varphi }(\xi )d\xi \end{aligned}$$
(A.2)
$$\begin{aligned} \mathbf {R}_{\mathbf {e}}^{\mathbf {DG}}= & {} \varvec{\varphi }(1)\varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }\left( \frac{1}{2}\right) - \int \limits _{\frac{1}{2}}^{1}\varvec{\varphi }'(\xi ) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.3)
$$\begin{aligned} \mathbf {L}_{\mathbf {e}}^{\mathbf {DG}}= & {} \varvec{\varphi }(0)\varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }\left( \frac{1}{2}\right) + \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }'(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) \varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.4)
$$\begin{aligned} \mathbf {R}_{\mathbf {p}}^{\mathbf {DG}}= & {} \varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }(0) + \int \limits _{\frac{1}{2}}^{1} \varvec{\varphi }(\xi )\varvec{\varphi }'\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.5)
$$\begin{aligned} \mathbf {L}_{\mathbf {p}}^{\mathbf {DG}}= & {} \varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }(1) - \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }(\xi )\varvec{\varphi }'\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.6)
$$\begin{aligned} \mathbf {M}_{\mathbf {L}}^{\mathbf {DG}}= & {} \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.7)
$$\begin{aligned} \mathbf {M}_{\mathbf {R}}^{\mathbf {DG}}= & {} \int \limits _{\frac{1}{2}}^1 \varvec{\varphi }(\xi )\varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.8)

1.2 Appendix A.2: Matrices and Tensors for the 1D Semi-implicit Sub-cell FV Method for \(\hbox {P}=2\)

$$\begin{aligned} \mathbf {R}^{\mathbf {FV}}= & {} \begin{pmatrix} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 5&{}0&{}0&{}0&{}0\\ -5&{}5&{}0&{}0&{}0\\ 0&{}-5&{}5&{}0&{}0\\ \end{pmatrix} \quad \mathbf {L}^{\mathbf {FV}}=\begin{pmatrix} 0&{}0&{}5&{}-5&{}0\\ 0&{}0&{}0&{}5&{}-5\\ 0&{}0&{}0&{}0&{}5\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ \end{pmatrix} \end{aligned}$$
(A.9)
$$\begin{aligned} \mathbf {M}_{\mathbf {R}}^{\mathbf {FV}}= & {} \begin{pmatrix} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0.5&{}0&{}0&{}0&{}0\\ 0.5&{}0.5&{}0&{}0&{}0\\ 0&{}0.5&{}0.5&{}0&{}0\\ \end{pmatrix} \quad \mathbf {M}_{\mathbf {L}}^{\mathbf {FV}}=\begin{pmatrix} 0&{}0&{}0.5&{}0.5&{}0\\ 0&{}0&{}0&{}0.5&{}0.5\\ 0&{}0&{}0&{}0&{}0.5\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ \end{pmatrix} \end{aligned}$$
(A.10)

1.3 Appendix A.3: Tensors for the Limited 1D Semi-implicit DG Method

$$\begin{aligned} \mathbf {R}^{\prime }{_{\mathbf {e}}^{\mathbf {DG}}}= & {} \varvec{\varphi }(1)\varvec{\varphi }\left( \frac{1}{2}\right) - \int \limits _{\frac{1}{2}}^{1}\varvec{\varphi }'(\xi ) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.11)
$$\begin{aligned} \mathbf {L}^{\prime }{_{\mathbf {e}}^{\mathbf {DG}}}= & {} \varvec{\varphi }(0)\varvec{\varphi }\left( \frac{1}{2}\right) + \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }'(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.12)

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Ioriatti, M., Dumbser, M. & Loubère, R. A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics. J Sci Comput 83, 27 (2020). https://doi.org/10.1007/s10915-020-01209-w

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  • DOI: https://doi.org/10.1007/s10915-020-01209-w

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