Anh-r-Adaptive Approximate Riemann Solver for the Euler Equations in Two Dimensions

1993 ◽  
Vol 14 (1) ◽  
pp. 185-217 ◽  
Author(s):  
Michael G. Edwards ◽  
J. Tinsley Oden ◽  
Leszek Demkowicz
Keyword(s):  
Euler Equations ◽  
Two Dimensions ◽  
Riemann Solver ◽  
Approximate Riemann Solver

scholarly journals An Approximate Riemann Solver for Euler Equations

10.5772/39028 ◽  
2012 ◽  
Author(s):  
Oscar Falcinelli ◽  
Sergio Elaskar ◽  
Jos Tamagno ◽  
Jorge Colman
Keyword(s):  
Euler Equations ◽  
Riemann Solver ◽  
Approximate Riemann Solver

A 3D Approximate Riemann Solver for the Euler Equations Using Flux Splitting Schemes With a Robust Multigrid

2016 ◽  
Author(s):  
Hong-Sik Im
Keyword(s):  
Euler Equations ◽  
Riemann Solver ◽  
Riemann Solvers ◽  
Approximate Riemann Solver ◽  
Time Stepping ◽  
Flux Vector Splitting ◽  
Time Marching ◽  
Convergence To Steady State ◽  
Flux Difference ◽  
Robust Multigrid

An explicit 3D approximate Riemann solver for the Euler equations is proposed using the famous shock capturing schemes with a simple cell vertex based multigrid method. A multistage Runge-Kutta time marching scheme with a local time stepping is used to achieve fast convergence to steady state. A Roe’s flux difference splitting, AUSM+, Van Leer and Steger-Warming’s flux vector splitting are implemented as base Riemann solvers with a third order flux reconstruction. It is shown that the proposed Riemann solvers accurately capture the shocks as well as reduce CPU time significantly with new multigrid.

scholarly journals A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows

2014 ◽  
Vol 15 (1) ◽  
pp. 46-75 ◽  
Author(s):  
J. Vides ◽  
B. Braconnier ◽  
E. Audit ◽  
C. Berthon ◽  
B. Nkonga
Keyword(s):  
Numerical Method ◽  
Euler Equations ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Source Term ◽  
Poisson Model ◽  
Riemann Solver ◽  
Approximate Riemann Solver ◽  
Relaxation Scheme ◽  
Astrophysical Flows

AbstractWe present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES [29] and several numerical experiments involving gravitational flows for astrophysics highlight the scheme.

The development of a Riemann solver for the steady supersonic Euler equations

1994 ◽  
Vol 98 (979) ◽  
pp. 325-339 ◽  
Author(s):  
E. F. Toro ◽  
A. Chakraborty
Keyword(s):  
Euler Equations ◽  
Weighted Average ◽  
Godunov Method ◽  
Riemann Solver ◽  
Flux Method ◽  
First Order ◽  
Enhanced Resolution ◽  
Average Flux ◽  
Exact Riemann Solver ◽  
Order Weighted Average

Abstract An improved version (HLLC) of the Harten, Lax, van Leer Riemann solver (HLL) for the steady supersonic Euler equations is presented. Unlike the HLL, the HLLC version admits the presence of the slip line in the structure of the solution. This leads to enhanced resolution of computed slip lines by Godunov type methods. We assess the HLLC solver in the context of the first order Godunov method and the second order weighted average flux method (WAF). It is shown that the improvement embodied in the HLLC solver over the HLL solver is virtually equivalent to incorporating the exact Riemann solver.

scholarly journals Approximate Riemann solver for hypervelocity flows

AIAA Journal ◽  
1992 ◽  
Vol 30 (10) ◽  
pp. 2558-2561 ◽  
Author(s):  
P. A. Jacobs
Keyword(s):  
Riemann Solver ◽  
Approximate Riemann Solver ◽  
Hypervelocity Flows

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

A Control-Volume Model of the Compressible Euler Equations with a Vertical Lagrangian Coordinate

2013 ◽  
Vol 141 (7) ◽  
pp. 2526-2544 ◽  
Author(s):  
Xi Chen ◽  
Natalia Andronova ◽  
Bram Van Leer ◽  
Joyce E. Penner ◽  
John P. Boyd ◽  
...  
Keyword(s):  
Euler Equations ◽  
Control Volume ◽  
Test Cases ◽  
Riemann Solver ◽  
New Approach ◽  
Vertical Coordinate ◽  
Low Mach Number ◽  
Atmospheric Flows ◽  
Lagrangian Coordinate ◽  
Control Volume Approach

Abstract Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.

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