The weighted average flux method applied to the Euler equations

1992 ◽  
Vol 341 (1662) ◽  
pp. 499-530 ◽  
Keyword(s):  
Euler Equations ◽  
Gas Dynamics ◽  
Hyperbolic Conservation Laws ◽  
Weighted Average ◽  
Two Dimensions ◽  
Test Problems ◽  
Flux Method ◽  
Hyperbolic Conservation ◽  
Equations Of Gas Dynamics ◽  
Average Flux

The weighted average flux method (WAF) for general hyperbolic conservation laws was formulated by Toro. Here the method is specialized to the time-dependent Euler equations of gas dynamics. Several improvements to the technique are presented. These have resulted from experience obtained from applying WAF to a variety of realistic problems. A hierarchy of solutions to the relevant Riemann problem, ranging from very simple approximations to the exact solution, are presented. Their performance in the WAF method for several test problems in one and two dimensions is assessed.

A linearized Riemann solver for the time-dependent Euler equations of gas dynamics

1991 ◽  
Vol 434 (1892) ◽  
pp. 683-693 ◽  
Keyword(s):  
Euler Equations ◽  
Gas Dynamics ◽  
Hyperbolic Conservation Laws ◽  
Weighted Average ◽  
Time Dependent ◽  
Test Problems ◽  
Riemann Solver ◽  
Equations Of Gas Dynamics ◽  
Exact Riemann Solver ◽  
Linearized Riemann Solver

The time-dependent Euler equations of gas dynamics are a set of nonlinear hyperbolic conservation laws that admit discontinuous solutions (e. g. shocks). In this paper we are concerned with Riemann-problem-based numerical methods for solving the general initial-value problem for these equations. We present an approximate, linearized Riemann solver for the time-dependent Euler equations. The solution is direct and involves few, and simple, arithmetic operations. The Riemann solver is then used, locally, in conjunction with the weighted average flux numerical method to solve the time-dependent Euler equations in one and two space dimensions with general initial data. For flows with shock waves of moderate strength the computed results are very accurate. For severe flow régimes we advocate the use of the present linearized Riemann solver in combination with the exact Riemann solver in an adaptive fashion. Numerical experiments demonstrate that such an approach can be very successful. One-dimensional and two-dimensional test problems show that the linearized Riemann solver is used in over 99% of the flow field producing net computing savings by a factor of about 2. A reliable and simple switching criterion is also presented. Results show that the adaptive approach effectively provides the resolution and robustness of the exact Riemann solver at the computing cost of the simple linearized Riemann solver. The relevance of the present methods concerns the numerical solution of multi-dimensional problems accurately and economically.

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.

A weighted average flux method for hyperbolic conservation laws

1989 ◽  
Vol 423 (1865) ◽  
pp. 401-418 ◽  
Keyword(s):  
Nonlinear Systems ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Model Equation ◽  
Hyperbolic Conservation Laws ◽  
Weighted Average ◽  
Second Order ◽  
Order Accuracy ◽  
Hyperbolic Conservation ◽  
Average Flux

A numerical technique, called a ‘weighted average flux’ (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving ‘generalized’ Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

A Very Efficient Class of HOC Schemes for the One-Dimensional Euler Equations of Gas Dynamics

2015 ◽  
Vol 813-814 ◽  
pp. 643-651 ◽  
Author(s):  
Bidyut B. Gogoi
Keyword(s):  
Shock Tube ◽  
Euler Equations ◽  
Gas Dynamics ◽  
Laval Nozzle ◽  
Numerical Solutions ◽  
Weighted Average ◽  
One Dimensional ◽  
Equations Of Gas Dynamics ◽  
De Laval Nozzle ◽  
The One

This manuscript introduces a class of higher order compact schemes for the solution of one dimensional (1-D) Euler equations of gas dynamics. These schemes are fourth order accurate in space and second or lower order accurate in time, depending on a weighted average parameter μ. The robustness and efficiency of our proposed schemes have been validated by applying them to three different shock-tube problems of gas dynamics, including the famous SOD shock-tube problem. Later on, the 1-D convergent-divergent nozzle problem (De laval nozzle problem) is also considered and numerical simulations are performed. In all the cases, our computed numerical solutions are found to be in excellent match with the exact solutions or available results in the existing literature. Overall the schemes are found to be efficient and accurate.

scholarly journals A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions

2014 ◽  
Vol 15 (4) ◽  
pp. 1184-1206 ◽  
Author(s):  
Zhenzhen Li ◽  
Xijun Yu ◽  
Jiang Zhu ◽  
Zupeng Jia
Keyword(s):  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Gas Dynamics ◽  
Two Dimensions ◽  
Oscillatory Property ◽  
Order Accuracy ◽  
Numerical Tests ◽  
Compressible Euler ◽  
Free Parameters ◽  
Runge Kutta

AbstractThis paper presents a new Lagrangian type scheme for solving the Euler equations of compressible gas dynamics. In this new scheme the system of equations is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh moves with the fluid flow. The scheme is conservative for the mass, momentum and total energy and maintains second-order accuracy. The scheme avoids solving the geometrical part and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.

scholarly journals SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH PRESCRIBED EIGENCURVES

2010 ◽  
Vol 07 (02) ◽  
pp. 211-254 ◽  
Author(s):  
HELGE KRISTIAN JENSSEN ◽  
IRINA A. KOGAN
Keyword(s):  
Conservation Laws ◽  
Gas Dynamics ◽  
Hyperbolic Conservation Laws ◽  
Space Dimension ◽  
Euler System ◽  
Complete Analysis ◽  
Overdetermined System ◽  
Hyperbolic Conservation ◽  
Algebraic Differential Equations ◽  
The Given

We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan–Kähler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class.

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