Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients.(English)Zbl 1078.35011

The authors study a model of conservative nonlinear conservation laws with a discontinuous flux function. More precisely, the following equation is being studied $$u_t + (k(x)u(1-u))_x = 0.$$ The authors find a particular entropy condition which has to be fullfiled on the line of discontinuity of the coefficient $$k$$, which guarantees the uniqueness of the entropy solution. Two new finite volume schemes are proposed to simulate the conservation laws with discontinuous coefficients. A comparison with commercial finite volume codes is done. It is shown that a loss of accuaracy of the commercial codes has been detected. A modified MUSCL method for stationary states is introduced as well.

MSC:

 35A35 Theoretical approximation in context of PDEs 35L65 Hyperbolic conservation laws 35R05 PDEs with low regular coefficients and/or low regular data 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76M12 Finite volume methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage
Full Text:

References:

 [1] DOI: 10.1016/S0045-7930(99)00026-2 · Zbl 0961.76048 [2] DOI: 10.1016/S0362-546X(97)00536-1 · Zbl 0989.35130 [3] DOI: 10.1007/978-3-662-22019-1 [4] DOI: 10.1137/S0036141093242533 · Zbl 0852.35094 [5] Dal Maso G., J. Math. Pures Appl. 74 pp 483– [6] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis VII (North-Holland, 2000) pp. 713–1020. [7] DOI: 10.1016/S0045-7930(02)00011-7 · Zbl 1084.76540 [8] DOI: 10.1002/fld.346 · Zbl 1053.76044 [9] DOI: 10.1007/978-1-4684-9486-0 [10] DOI: 10.1137/0733001 · Zbl 0876.65064 [11] Godunov S. K., Mat. Sb. 47 pp 271– [12] DOI: 10.1007/978-1-4612-0713-9 [13] DOI: 10.1016/0021-9991(83)90066-9 · Zbl 0565.65049 [14] DOI: 10.1137/0148059 · Zbl 0688.35056 [15] DOI: 10.1016/0022-0396(86)90037-9 · Zbl 0612.35085 [16] DOI: 10.1137/0148079 · Zbl 0688.35057 [17] DOI: 10.1090/conm/108/1068333 [18] DOI: 10.1006/jdeq.1998.3624 · Zbl 0935.35097 [19] DOI: 10.1080/03605309508821159 · Zbl 0836.35090 [20] DOI: 10.1006/jdeq.2000.3826 · Zbl 0977.35083 [21] Kružkov S. N., Mat. Sb. 81 pp 228– [22] DOI: 10.1137/0732038 · Zbl 0830.35079 [23] Lin L. W., SIAM J. Numer. Anal. 32 pp 841– [24] DOI: 10.1016/S0196-8858(82)80010-9 · Zbl 0508.76107 [25] DOI: 10.1137/S0036142999363668 · Zbl 0972.65060 [26] DOI: 10.1137/S0036142900374974 · Zbl 1055.65104 [27] DOI: 10.1007/s002050100157 · Zbl 0999.35018 [28] DOI: 10.1016/0021-9991(79)90145-1 · Zbl 1364.65223 [29] Vol’pert A. I., Mat. Sb. 73 pp 255–
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.