Efficient implementation of essentially nonoscillatory shock-capturing schemes. (English) Zbl 0653.65072

Essentially nonoscillatory (ENO) schemes for nonlinear hyperbolic systems have the disadvantage of being very difficult to implement, particularly in more than one spatial dimension or when reaction terms are present in the equation. A new class of ENO schemes is introduced which is much easier to program. The methods use Runge-Kutta time discretizations, and for the representation of the fluxes no cell averages are required. Application of the methods to some standard problems indicates that they work very well in regions near shocks and where the solution is smooth. Near contact discontinuities, however, the methods are excessively diffusive.
Reviewer: G.Hedstrom


65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
35L60 First-order nonlinear hyperbolic equations
Full Text: DOI


[1] Gear, C.W., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0217.21701
[2] Harten, A., J. comput. phys., 49, 357, (1983)
[3] Harten, A., SIAM J. numer. anal., 21, 1, (1984)
[4] Harten, A., ()
[5] Harten, A.; Osher, S., SIAM J. numer. anal., 24, 279, (1987)
[6] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., J. comput. phys., 71, 231, (1987)
[7] Lambert, J.D., Computational methods in ordinary differential equations, (1973), Wiley New York · Zbl 0258.65069
[8] Lax, P.D.; Wendroff, B., Commun. pure appl. math., 13, 217, (1960)
[9] Osher, S.; Chakravarthy, S., SIAM J. numer. anal., 21, 955, (1984)
[10] Osher, S.; Chakravarthy, S., (), 229
[11] Roe, P., J. comput. phys., 43, 357, (1981)
[12] {\scR. Sanders}, Math. Comput., in press.
[13] {\scC. Shu}, TVB Math. Comp., in press.
[14] {\scC. Shu}, TVD time discretizations, preprint.
[15] Sweby, P.K., SIAM J. numer. anal., 21, 995, (1984)
[16] Van Leer, B., J. comput. phys., 14, 361, (1974)
[17] Van Leer, B., J. comput. phys., 32, 101, (1979)
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.