First order schemes for the Euler equations

2011 ◽  
pp. 39-64
Keyword(s):  
Euler Equations ◽  
First Order

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 On the Impossibility of First-Order Phase Transitions in Systems Modeled by the Full Euler Equations

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1039
Author(s):  
Maren Hantke ◽  
Ferdinand Thein
Keyword(s):  
Euler Equations ◽  
Single Phase ◽  
Two Phase Flows ◽  
Two Phase ◽  
First Order ◽  
Engineering Sciences ◽  
The One ◽  
First Order Phase ◽  
Isothermal Euler Equations ◽  
Appl Math

Liquid–vapor flows exhibiting phase transition, including phase creation in single-phase flows, are of high interest in mathematics, as well as in the engineering sciences. In two preceding articles the authors showed on the one hand the capability of the isothermal Euler equations to describe such phenomena (Hantke and Thein, arXiv, 2017, arXiv:1703.09431). On the other hand they proved the nonexistence of certain phase creation phenomena in flows governed by the full system of Euler equations, see Hantke and Thein, Quart. Appl. Math. 2015, 73, 575–591. In this note, the authors close the gap for two-phase flows by showing that the two-phase flows considered are not possible when the flow is governed by the full Euler equations, together with the regular Rankine-Hugoniot conditions. The arguments rely on the fact that for (regular) fluids, the differences of the entropy and the enthalpy between the liquid and the vapor phase of a single substance have a strict sign below the critical point.

Effect of Dissipative Terms on the Quality of Two and Three-Dimensional Euler Flow Solutions

2008 ◽  
Author(s):  
Seyed Saied Bahrainian
Keyword(s):  
Euler Equations ◽  
Hyperbolic Conservation Laws ◽  
Three Dimensional ◽  
Strong Shock ◽  
Computational Domain ◽  
Third Order ◽  
First Order ◽  
Proper Values ◽  
Dissipative Terms

The Euler equations are a set of non-dissipative hyperbolic conservation laws that can become unstable near regions of severe pressure variation. To prevent oscillations near shockwaves, these equations require artificial dissipation terms to be added to the discretized equations. A combination of first-order and third-order dissipative terms control the stability of the flow solutions. The assigned magnitude of these dissipative terms can have a direct effect on the quality of the flow solution. To examine these effects, subsonic and transonic solutions of the Euler equations for a flow passed a circular cylinder has been investigated. Triangular and tetrahedral unstructured grids were employed to discretize the computational domain. Unsteady Euler equations are then marched through time to reach a steady solution using a modified Runge-Kutta scheme. Optimal values of the dissipative terms were investigated for several flow conditions. For example, at a free stream Mach number of 0.45 strong shock waves were captured on the cylinder by using values of 0.25 and 0.0039 for the first-order and third-order dissipative terms. In addition to the shock capturing effect, it has been shown that smooth pressure coefficients can be obtained with the proper values for the dissipative terms.

scholarly journals Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

2011 ◽  
Vol 18 (3) ◽  
pp. 351-358 ◽  
Author(s):  
M. Dunphy ◽  
C. Subich ◽  
M. Stastna
Keyword(s):  
Euler Equations ◽  
Solitary Waves ◽  
Dynamic Simulations ◽  
Large Lakes ◽  
Full Spectrum ◽  
Density Profiles ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
First Order ◽  
Spectral And Pseudospectral Methods

Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

scholarly journals Comparative study of numerical schemes for strong shock simulation using the Euler equations

2015 ◽  
Vol 18 (1) ◽  
pp. 73-88
Author(s):  
Binh Huy Nguyen ◽  
Giang Song Le
Keyword(s):  
Euler Equations ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Strong Shock ◽  
Strong Discontinuity ◽  
Numerical Schemes ◽  
First Order ◽  
Flux Vector Splitting ◽  
Explicit Finite Difference ◽  
Eno Scheme

A numerical study of extremely strong shocks was presented. Various types of numerical schemes with first-order accuracy and higherorder accuracy with adaptive stencils were implemented to solve the one and twodimensional Euler equations based on the explicit finite difference method, including Roe’s first-order upwind, Steger-Warming Flux Vector splitting (FVS), Sweby’s flux-limited and Essentially Non-oscillatory (ENO) scheme. The result comparisons were carried out to discuss which scheme is the most suitable for strong shock problem. The dissipative nature of the firstorder scheme can be easily seen from the numerical solutions. High order ENO scheme had the best resolution for the case having weak discontinuity, but it over- predicted the shock wave location for the case of strong discontinuity.

scholarly journals Causal dissipation for the relativistic dynamics of ideal gases

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160729 ◽  
Author(s):  
Heinrich Freistühler ◽  
Blake Temple
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Second Order ◽  
Order System ◽  
Navier Stokes ◽  
Unique Element ◽  
Relativistic Euler Equations ◽  
Navier Stokes Equations ◽  
First Order ◽  
Ideal Gases

We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.

A WELL-BALANCED SCHEME USING NON-CONSERVATIVE PRODUCTS DESIGNED FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS WITH SOURCE TERMS

2001 ◽  
Vol 11 (02) ◽  
pp. 339-365 ◽  
Author(s):  
LAURENT GOSSE
Keyword(s):  
Asymptotic Behavior ◽  
Conservation Laws ◽  
Euler Equations ◽  
Hyperbolic Systems ◽  
Zero Order ◽  
Inhomogeneous System ◽  
Source Terms ◽  
First Order ◽  
Systems Of Conservation Laws ◽  
The Right

The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a non-conservative reformulation of the zero-order terms of the right-hand side of the equations. In this context, we decided to use the results of Dal Maso, Le Floch and Murat about non-conservative products, and the generalized Roe matrices introduced by Toumi to derive a first-order linearized well-balanced scheme in the sense of Greenberg and Le Roux. As a main feature, this approach is able to preserve the right asymptotic behavior of the original inhomogeneous system, which is not an obvious property. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations.

Long-time shallow-water equations with a varying bottom

1997 ◽  
Vol 349 ◽  
pp. 173-189 ◽  
Author(s):  
ROBERTO CAMASSA ◽  
DARRYL D. HOLM ◽  
C. DAVID LEVERMORE
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Euler Equations ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Three Dimensional ◽  
Local Conservation ◽  
First Order ◽  
Long Time ◽  
Conservation Laws Of Energy

We present and discuss new shallow-water equations that model the long-time effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid possessing a free surface and moving under the force of gravity. We consider the regime where the Froude number ε is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained from the ε→0 limit of the Euler equations (the rigid-lid approximation) at the first order of an asymptotic expansion in δ2. These equations possess local conservation laws of energy and vorticity which reflect exact layer mean conservation laws of the three-dimensional Euler equations. In addition, they convect potential vorticity and have a Hamilton's principle formulation. We contrast them with the Green–Naghdi equations.

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations

2016 ◽  
Vol 8 (4) ◽  
pp. 670-692 ◽  
Author(s):  
Huajun Zhu ◽  
Xiaogang Deng ◽  
Meiliang Mao ◽  
Huayong Liu ◽  
Guohua Tu
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Mach Reflection ◽  
Finite Volume Methods ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Shock Stability ◽  
Double Mach Reflection ◽  
High Mach

AbstractWe compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: A first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.

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