A linearized Riemann solver for the steady supersonic Euler equations

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.

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

The Aeronautical Journal ◽

10.1017/s0001924000026890 ◽

1994 ◽

Vol 98 (979) ◽

pp. 325-339 ◽

Cited By ~ 16

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.

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An auto-adaptive approximate Riemann solver for non-linear Euler equations

Computers & Fluids ◽

10.1016/j.compfluid.2012.07.001 ◽

2012 ◽

Vol 68 ◽

pp. 219-243

Author(s):

Capdeville Guy

Keyword(s):

Euler Equations ◽

Riemann Solver ◽

Approximate Riemann Solver ◽

Non Linear

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

Monthly Weather Review ◽

10.1175/mwr-d-12-00129.1 ◽

2013 ◽

Vol 141 (7) ◽

pp. 2526-2544 ◽

Cited By ~ 12

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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