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Volume 28, Issue 2
An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations

Andrea Thomann, Markus Zenk, Gabriella Puppo & Christian Klingenberg

DOI: 10.4208/cicp.OA-2019-0123

Commun. Comput. Phys., 28 (2020), pp. 591-620.

Published online: 2020-06

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  • Abstract

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.

  • Keywords

Finite volume methods, Euler equations, positivity preserving, asymptotic preserving, relaxation, low Mach scheme, IMEX schemes.

  • AMS Subject Headings

76M12, 76M45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-591, author = {Thomann , AndreaZenk , MarkusPuppo , Gabriella and Klingenberg , Christian}, title = {An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {2}, pages = {591--620}, abstract = {

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0123}, url = {http://global-sci.org/intro/article_detail/cicp/16942.html} }
TY - JOUR T1 - An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations AU - Thomann , Andrea AU - Zenk , Markus AU - Puppo , Gabriella AU - Klingenberg , Christian JO - Communications in Computational Physics VL - 2 SP - 591 EP - 620 PY - 2020 DA - 2020/06 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0123 UR - https://global-sci.org/intro/article_detail/cicp/16942.html KW - Finite volume methods, Euler equations, positivity preserving, asymptotic preserving, relaxation, low Mach scheme, IMEX schemes. AB -

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.

Thomann , AndreaZenk , MarkusPuppo , Gabriella and Klingenberg , Christian. (2020). An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations. Communications in Computational Physics. 28 (2). 591-620. doi:10.4208/cicp.OA-2019-0123
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