Implicit Solution of the Unsteady Euler Equations for High-Order Accurate Discontinuous Galerkin Discretizations

2006 ◽  
Author(s):  
Li Wang ◽  
Dimitri Mavriplis
Keyword(s):  
Euler Equations ◽  
Discontinuous Galerkin ◽  
High Order

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.

scholarly journals A Comparison between High-order Temporal Integration Methods Applied to the Discontinuous Galerkin Discretized Euler Equations

2014 ◽  
Vol 45 ◽  
pp. 518-527 ◽  
Author(s):  
Alessandra Nigro ◽  
Carmine De Bartolo ◽  
Salvatore M. Renda ◽  
Francesco Bassi
Keyword(s):  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Temporal Integration ◽  
High Order ◽  
Integration Methods

scholarly journals Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes

2021 ◽  
Author(s):  
Hendrik Ranocha ◽  
Gregor J. Gassner
Keyword(s):  
Linear Stability ◽  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Density Wave ◽  
High Order ◽  
Local Linear ◽  
Compressible Euler ◽  
Pressure Equilibrium ◽  
Stability Issues ◽  
Split Form

AbstractRecently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (10.5281/zenodo.4054366).

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