A study of numerical methods for hyperbolic conservation laws with stiff source terms. (English) Zbl 0682.76053

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a model advection equation with a parameter-dependent source term is studied. Two approaches to incorporate the source terms are utilized: MacCormack type predictor-corrector methods with flux limiters and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Comparisons over a wide range of parameter values are made. On the whole, the splitting methods perform somewhat better. In the stiff case, a numerical phenomenon of incorrect propagation speeds of discontinuities is observed and explained. Using the model scalar equation, we show that this is due to the introduction of nonequilibrium values through numerical dissipation in the advection step.


76N15 Gas dynamics (general theory)
76V05 Reaction effects in flows
76K05 Hypersonic flows
35L65 Hyperbolic conservation laws
Full Text: DOI


[1] Aki, T., National aerospace laboratory technical report, (1987), (unpublished)
[2] Boris, J.P.; Oran, E.S., Numerical simulation of reactive plow, (1987), Elsevier Amsterdam/New York
[3] Bussing, T.R.A.; Murman, E.M., AIAA paper 850331, (1985), (unpublished)
[4] Carofano, G.C., Technical report ARLCB-TR-84029, (1984), (unpublished)
[5] Colella, P.; Majda, A.; Roytburd, V., SIAM J. sci. stat. comput., 7, 1059, (1986)
[6] Colella, P.; Woodward, P., J. comput. phys., 54, 174, (1984)
[7] Drummond, J.P.; Rogers, R.C.; Hussaini, M.Y., AIAA paper 861327, (1986), (unpublished)
[8] Gear, C.W., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0217.21701
[9] {\scA. Harten}, pivate communication (1987).
[10] Harten, A., (), (unpublished)
[11] Lambert, J.D., Computational methods in ordinary differential equations, (1973), Wiley New York · Zbl 0258.65069
[12] Lax, P.D.; Wendroff, B., Commun pure appl. math., 13, 217, (1960)
[13] Lee, J., AIAA paper 841729, (1984)
[14] MacCormack, R.W., AIAA paper 69354, (1969)
[15] Roe, P.L., J. comput. phys., 43, 357, (1981)
[16] Sandham, N.D.; Yee, H.C., NASA technical memorandum 102194, (1989), (unpublished)
[17] Strang, G., SIAM J. num. anal., 5, 506, (1968)
[18] Vincenti, W.G.; Kruger, C.H., Introduction to physical gas dynamics, (1967), Wiley New York
[19] Warming, R.F.; Kutler, P.; Lomax, H., Aiaa j., 189, (1973)
[20] Yee, H.C., NASA ames technical memoranda 89464, (1987), (unpublished), and 101088, 1989 (unpublished)
[21] Yee, H.C.; Shinn, J.L., AIAA paper 871116, (1987), (unpublished)
[22] Young, T.R.; Boris, J.P., J. phys. chem., 81, 2424, (1977)
[23] Young, V.Y.C.; Yee, H.C., AIAA paper 870112, (1987), (unpublished)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.