Synonyms
Flux function
Definition
Given a set of hyperbolic conservation laws in one space dimension plus time, the Riemann problem is to find the solution to the special initial value problem in which two different constant states each occupy one half of the initial line. It is an essential building block in many versions of computational fluid dynamics.
Overview
To understand the definition, it is helpful to consider a specific example of a Riemann problem, namely, the shock tube problem. This involves a common experiment in gas dynamics in which a tube is divided into left and right parts, separated by a diaphragm. One half of the tube is filled with a gas at high pressure and the other half with a gas at low pressure. At some moment, the diaphragm is ruptured, and the high-pressure gas rushes toward the low-pressure gas, pushing it ahead at high speed. In the case of a general Riemann problem, however, the left and right states are entirely arbitrary (there is no requirement that...
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References
Berthon, C., Cocquel, F., LeFloch, P.: Why many theories of shock waves are necessary: kinetic relations for nonconservative systems (2011). arXiv:1006.1102v2
Bultelle, M., Grassin, M., Serre, D.: Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal. 35(6), 2272–2297 (1998)
Einfeldt, B., Munz C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92(2), 273–295 (1991). doi:10.1016/0021-9991(91)90211-3
Elling, V.: The carbuncle phenomenon is incurable. Acta Math. Sci. 29(6), 1647–1656 (2009)
Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 3, 271–306 (1959)
Harten, A., Lax, P.D., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)
Jameson, A.: Analysis and design of numerical schemes for gas dynamics, 2: artificial diffusion and discrete shock structure. Int. J. Comput. Fluid Dyn. 5(1), 1–29 (1995)
Linde T.: A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws. Int. J. Numer. Methods Fluids 40, 391–402 (2002)
Liou, M.-S.: A Sequel to AUSM, part II: AUSM+-up. J. Comput. Phys. 214, 137–170 (2006)
Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)
Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 49(1), 25–34 (1994)
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Roe, P.L. (2015). Riemann Problem. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_357
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DOI: https://doi.org/10.1007/978-3-540-70529-1_357
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