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Unsplit algorithms for multidimensional systems of hyperbolic conservation laws with source terms. (English) Zbl 0999.65089
Summary: This work describes an unsplit, second-order accurate algorithm for multidimensional systems of hyperbolic conservation laws with source terms, such as the compressible Euler equations for reacting flows. It is a MUSCL-type, shock-capturing scheme that integrates all terms of the governing equations simultaneously, in a single time-step, thus avoiding dimensional or time-splitting. Appropriate families of space-time manifolds are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding 1-D homogeneous system. The local geometry, of these manifolds depends on the source terms and the spatial derivatives of the flow variables.
Numerical integration of the characteristic equations is performed along these manifolds in the upwinding part of the algorithm. Numerical simulations of two-dimensional detonations with simplified kinetics are performed to test the accuracy and robustness of the algorithm. These flows are unstable for a wide range of parameters and may exhibit chaotic behavior. Grid-convergence studies and comparisons with earlier results, obtained with traditional schemes, are presented.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics, general
76L05 Shock waves and blast waves in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
80A25 Combustion
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
Full Text: DOI
[1] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 35, (1965) · Zbl 0141.28902
[2] Lax, P., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, (1973), SIAM · Zbl 0268.35062
[3] van Leer, B., Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method, J. comp. phys., 32, 101, (1979) · Zbl 1364.65223
[4] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate ENO schemes, III, J. comp. phys., 71, 231, (1987) · Zbl 0652.65067
[5] Colella, P.; Woodward, P.R., (), 174
[6] Deconick, H.; Hirsch, C.; Peuteman, J., Characteristic decomposition methods for the multidimensional Euler equations, 10^th international conference on numerical methods bejing in fluid dynamics, AIAA paper no. 89-1958, (1986)
[7] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comp. phys., 87, 171, (1988) · Zbl 0694.65041
[8] Leveque, L.J., High resolution finite volume methods on arbitrary grids via wave propagation, J. comp. phys., 78, 33, (1988)
[9] Dai, W.L.; Woodward, P.R., A 2^nd order unsplit Godunov scheme for 2-dimensional and 3-dimensional Euler equations, J. comp. phys., 134, 261, (1997)
[10] Lappas, T.; Leonard, A.; Dimotakis, P.E., Riemann invariant manifolds for the multidimensional Euler equations, SIAM J. sci. comp., 20, 1481, (1999) · Zbl 0986.76078
[11] Papalexandris, M.V.; Leonard, A.; Dimotakis, P.E., Unsplit schemes for hyperbolic conservation laws with source terms in one space dimension, J. comp. phys., 134, 31, (1997) · Zbl 0880.65074
[12] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, engquist-osher and roe, SIAM J. sci. stat. comp., 5, 1, (1984) · Zbl 0547.65065
[13] Ben-Artzi, M., The generalized Riemann problem for reactive flows, J. comp. phys., 81, 744, (1989)
[14] Fickett, W.; Davis, W.C., Detonation, (1979), U.C. Berkeley Press
[15] Erpenbeck, J.J., Stability of idealized one-reaction detonations, Phys. fluids, 7, 684, (1964) · Zbl 0123.42901
[16] Yao, J.; Stewart, J.S., On the dynamics of multi-dimensional detonations, J. fl. mech., 309, 225, (1996)
[17] Clavin, P.; He, L.; Williams, F.A., Multi-dimensional stability analysis of overdriven gaseous detonations, Phys. fl., 9, 3764-3785, (1997)
[18] Taki, S.; Fujiwara, T., Numerical analysis of two-dimensional non-steady detonations, AIAA journal, 16, 73, (1973)
[19] Oran, E.S.; Kailasanath, K.; Guirguis, R.H., Numerical simulations of the development and structure of detonations, Prog. aeronaut. astronaut., 114, 155, (1988)
[20] Bourlioux, A.; Majda, A.J., Theoretical and numerical structure for unstable two-dimensional detonations, Combust. flame, 90, 211, (1992)
[21] Cai, W., High-order hybrid numerical simulations of 2-dimensional detonation waves, AIAA journal, 33, 1248, (1995) · Zbl 0844.76070
[22] Pratt, D.T.; Humphrey, J.W.; Glenn, D.E., Morphology of standing oblique detonation waves, J. prop. power, 7, 837, (1991)
[23] Li, C.; Kailasanath, K.; Oran, E.S., Detonation structures behind oblique shocks, Phys. fluids, 6, 1600, (1994) · Zbl 0825.76396
[24] Papalexandris, M.V., Numerical study of wedge-induced detonations, Combustion and flame, 120, 526, (2000)
[25] Barthel, H.O., Reaction zone—shock front coupling in detonations, Phys. fluids, 15, 43, (1972)
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