A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics
| dc.contributor.author | IORIATTI, Matteo | |
| hal.structure.identifier | Department of civil, environmental and mechanical engineering [Trento] | |
| dc.contributor.author | DUMBSER, Michael | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | LOUBÈRE, Raphaël | |
| dc.date.accessioned | 2024-04-04T02:47:53Z | |
| dc.date.available | 2024-04-04T02:47:53Z | |
| dc.date.issued | 2020-05 | |
| dc.identifier.issn | 0885-7474 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191688 | |
| dc.description.abstractEn | 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 (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 (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. The resulting pressure system 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. in 2014 for explicit DG schemes on collocated grids and is based on the a posteriori MOOD approach of Clain, Loubère and Diot. Recently, this approach was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in [44]. 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 cells, 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 any non-physical spurious numerical oscillations. | |
| dc.language.iso | en | |
| dc.publisher | Springer Verlag | |
| dc.subject.en | high order semi-implicit discontinuous Galerkin schemes | |
| dc.subject.en | a posteriori sub-cell finite volume limiter | |
| dc.subject.en | staggered grids | |
| dc.subject.en | MOOD paradigm | |
| dc.subject.en | sub-cell finite volume limiting | |
| dc.subject.en | Euler equations of compressible gasdynamics | |
| dc.title.en | A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1007/s10915-020-01209-w | |
| dc.subject.hal | Mathématiques [math] | |
| bordeaux.journal | Journal of Scientific Computing | |
| bordeaux.volume | 83 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 2 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-02618986 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02618986v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Scientific%20Computing&rft.date=2020-05&rft.volume=83&rft.issue=2&rft.eissn=0885-7474&rft.issn=0885-7474&rft.au=IORIATTI,%20Matteo&DUMBSER,%20Michael&LOUB%C3%88RE,%20Rapha%C3%ABl&rft.genre=article |
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