A NEW APPROXIMATE RIEMANN SOLVER APPLIED TO HLLC METHOD

A NEW APPROXIMATE RIEMANN SOLVER APPLIED TO HLLC METHOD

Elaskar, Sergio; Falcinelli, Oscar; Tamagno, José

Full Article:

In solving Euler equations applying finite volume techniques, the calculations of numerical fluxes across cell interfaces, have become an essential item. The numerical scheme exactitude, the ability of handle discontinuities and the correct prediction of the propagating waves velocity, are strongly dependent on such numerical fluxes. The pioneer work of Godunov[1] was the starting point to solve the Euler equations by means of Riemann solvers. The excellent results obtained with Godunov technique, motivated several lines of research with the purpose of extending it to three dimensional flows, to achieve higher order of accuracy, etc. All calculating schemes that incorporate Riemann solvers are very precise, but unfortunately, computational demands are intense because of the non linear system of algebraic equations which must be solved in an iterative manner. An alternative which will demand less computational effort, could be provided by the use of approximate Riemann solvers, although less accurate and also, less robust. In this paper, an approximate Riemann solver which does not require iterations, possesses a high degree of accuracy and a lower computational demand in solving the Euler equations, is described. It is based on the use of dimensional analysis to reduce the number of independent variables needed to outline the physics of the problem. The scheme here presented is compared in accuracy as well as in computational effort with an exact iterative solver and with three well known approximated solvers: the Two Rarefactions Riemann Solver, the Two Shocks Riemann Solver, and an Adaptive version of these two. Substantially smaller mean errors have been found with the approximation here presented than those found with the best of all the above mentioned approximated solvers. Finally, a finite volume computer code to solve one-dimensional Euler equations using the Harten, Lax and van Leer Contact (HLLC) scheme, was developed. Results obtained solving the Shock Tube problem with the HLLC scheme, have shown no significant differences in accuracy and robustness when either the new approximate Riemann solver or the exact solver, are used. From the point of view of computer resources, the new approximate solver offers advantages.

In solving Euler equations applying finite volume techniques, the calculations of numerical fluxes across cell interfaces, have become an essential item. The numerical scheme exactitude, the ability of handle discontinuities and the correct prediction of the propagating waves velocity, are strongly dependent on such numerical fluxes. The pioneer work of Godunov[1] was the starting point to solve the Euler equations by means of Riemann solvers. The excellent results obtained with Godunov technique, motivated several lines of research with the purpose of extending it to three dimensional flows, to achieve higher order of accuracy, etc. All calculating schemes that incorporate Riemann solvers are very precise, but unfortunately, computational demands are intense because of the non linear system of algebraic equations which must be solved in an iterative manner. An alternative which will demand less computational effort, could be provided by the use of approximate Riemann solvers, although less accurate and also, less robust. In this paper, an approximate Riemann solver which does not require iterations, possesses a high degree of accuracy and a lower computational demand in solving the Euler equations, is described. It is based on the use of dimensional analysis to reduce the number of independent variables needed to outline the physics of the problem. The scheme here presented is compared in accuracy as well as in computational effort with an exact iterative solver and with three well known approximated solvers: the Two Rarefactions Riemann Solver, the Two Shocks Riemann Solver, and an Adaptive version of these two. Substantially smaller mean errors have been found with the approximation here presented than those found with the best of all the above mentioned approximated solvers. Finally, a finite volume computer code to solve one-dimensional Euler equations using the Harten, Lax and van Leer Contact (HLLC) scheme, was developed. Results obtained solving the Shock Tube problem with the HLLC scheme, have shown no significant differences in accuracy and robustness when either the new approximate Riemann solver or the exact solver, are used. From the point of view of computer resources, the new approximate solver offers advantages.

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DOI: 10.5151/meceng-wccm2012-18534

Referências bibliográficas

• [1] Godunov, S. (1959). A Finite Difference Method for the computation of Discontinuous Solutions of the Equations of Fluid Dynamics. Mat. Sb., Vol. 47, pp. 357-393 (In Russian).
• [2] Toro, E. (2009). Riemann Solvers and Numerical Methods for Fluid Mechanics. A practical introduction (Third Edition), Springer-Verlag, ISBN 978-3-540-25202-3, Berlin.
• [3] LeVeque, R. (2004). Finite Volume Methods for Hyperbolic Problems (Second Edition), Cambridge University Press, ISBN 0-521-00924-3, Cambridge.
• [4] Yee, H. (1989) A Class of High Resolution Explicit and Implicit Shock-Capturing Methods. NASA Technical memorandum 101088. Ames Research Center, California.
• [5] Toro, E., Spruse, M. and Speares, W. (1994). Restoration of the Contact Surface in the HLL-Riemann solver. Shock Waves, Vol. 4, pp. 25-34, ISSN 0938-1287.
• [6] van Leer, B. (1985). On the Relation Between the Upwind-Differencing Schemes of Godunov, Enguist-Osher and Roe. SIAM Journal of Scientific Computing, Vol. 5, No. 1, pp. 1-20, ISSN 1064-8275.

Como citar:

Elaskar, Sergio; Falcinelli, Oscar; Tamagno, José; "A NEW APPROXIMATE RIEMANN SOLVER APPLIED TO HLLC METHOD", p-1754-1769. In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 23580828, DOI 10.5151/meceng-wccm2012-18534