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.

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.

Robust HLLC Riemann solver with weighted average flux scheme for strong shock

2009 ◽  
Vol 228 (20) ◽  
pp. 7634-7642 ◽  
Author(s):  
Sung Don Kim ◽  
Bok Jik Lee ◽  
Hyoung Jin Lee ◽  
In-Seuck Jeung
Keyword(s):  
Strong Shock ◽  
Weighted Average ◽  
Riemann Solver ◽  
Average Flux

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

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