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High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics


Abgrall, Rémi; Bacigaluppi, Paola; Tokareva, Svetlana (2019). High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics. Computers & Mathematics with Applications, 78(2):274-297.

Abstract

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation (Ricchiuto and Abgrall, 2010; Abgrall, 2006) with a Deferred Correction (DeC) type method (Liu et al., 2008; Minion, 2003), allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of Abgrall et al. (2016) and Abgrall (2017) to multidimensional systems. We have assessed our method on several challenging benchmark problems for one- and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Abstract

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation (Ricchiuto and Abgrall, 2010; Abgrall, 2006) with a Deferred Correction (DeC) type method (Liu et al., 2008; Minion, 2003), allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of Abgrall et al. (2016) and Abgrall (2017) to multidimensional systems. We have assessed our method on several challenging benchmark problems for one- and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

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Additional indexing

Item Type:Journal Article, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Modeling and Simulation
Physical Sciences > Computational Theory and Mathematics
Physical Sciences > Computational Mathematics
Uncontrolled Keywords:Modelling and Simulation, Computational Theory and Mathematics, Computational Mathematics
Language:English
Date:1 July 2019
Deposited On:01 Nov 2018 11:38
Last Modified:29 Jul 2020 07:58
Publisher:Elsevier
ISSN:0898-1221
OA Status:Green
Publisher DOI:https://doi.org/10.1016/j.camwa.2018.05.009
Project Information:
  • : FunderSNSF
  • : Grant ID200021_153604
  • : Project TitleHigh fidelity simulation for compressible material

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