Implicit finite element schemes for the stationary compressible Euler equations

2011 ◽  
Vol 69 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Marcel Gurris ◽  
Dmitri Kuzmin ◽  
Stefan Turek
Keyword(s):  
Finite Element ◽  
Euler Equations ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Finite Element Schemes

scholarly journals Efficient Parallel 3D Computation of the Compressible Euler Equations with an Invariant-domain Preserving Second-order Finite-element Scheme

2021 ◽  
Vol 8 (3) ◽  
pp. 1-30
Author(s):  
Matthias Maier ◽  
Martin Kronbichler
Keyword(s):  
Finite Element ◽  
Euler Equations ◽  
High Performance ◽  
Ad Hoc ◽  
Second Order ◽  
Compressible Euler Equations ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Invariant Domain ◽  
Compressible Euler

We discuss the efficient implementation of a high-performance second-order collocation-type finite-element scheme for solving the compressible Euler equations of gas dynamics on unstructured meshes. The solver is based on the convex-limiting technique introduced by Guermond et al. (SIAM J. Sci. Comput. 40, A3211–A3239, 2018). As such, it is invariant-domain preserving ; i.e., the solver maintains important physical invariants and is guaranteed to be stable without the use of ad hoc tuning parameters. This stability comes at the expense of a significantly more involved algorithmic structure that renders conventional high-performance discretizations challenging. We develop an algorithmic design that allows SIMD vectorization of the compute kernel, identify the main ingredients for a good node-level performance, and report excellent weak and strong scaling of a hybrid thread/MPI parallelization.

Numerical Solution of Compressible Euler Equations by High Order Nodal Discontinuous Galerkin Method

2013 ◽  
Vol 392 ◽  
pp. 165-169 ◽  
Author(s):  
Fareed Ahmed ◽  
Faheem Ahmed ◽  
Yong Yang
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Gas Dynamics ◽  
High Order ◽  
Compressible Euler Equations ◽  
Element Space ◽  
Numerical Predictions ◽  
Compressible Euler

In this paper we present a robust, high order method for numerical solution of compressible Euler Equations of the gas dynamics. Euler equations are hyperbolic in nature. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method combines mainly two key ideas which are based on the finite volume and finite element methods. In this method, we employ Discontinuous Galerkin (DG) technique for finite element space discretization by discontinuous approximations. Whereas, for temporal discretization, we used explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. We used filter to remove spurious oscillations near the shock waves. Numerical predictions for Shock tube problem (SOD) are presented and compared with exact solution at different polynomial order and mesh sizes. Results show the suitability of DG method for modeling gas dynamics equations and effectiveness of high order approximations.

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