×

zbMATH — the first resource for mathematics

Stable numerical methods for conservation laws with discontinuous flux function. (English) Zbl 1088.65080
Summary: We develop numerical methods for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients. Using a relaxation approximation, the nonlinear equation is transformed to a semilinear diagonalizable problem with linear characteristic variables. Eulerian and Lagrangian methods are used for the advection stage while an implicit-explicit scheme solves the relaxation stage. The main advantages of this approach are neither Riemann problem solvers nor linear iterations are required during the solution process. Moreover, the characteristic-based relaxation method is unconditionally stable such that no CFL conditions are imposed on the selection of time steps. Numerical results are shown for models on traffic flows and two-phase flows.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
90B20 Traffic problems in operations research
76T10 Liquid-gas two-phase flows, bubbly flows
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Banda, M. Seaïd, Higher-order relaxation schemes for hyperbolic systems of conservation laws, J. Numer. Math. (in press). · Zbl 1084.65076
[2] Banda, M.; Seaïd, M.; Klar, A.; Pareschi, L., Compressible and incompressible limits for hyperbolic systems with relaxation, J. comp. appl. math., 168, 41-52, (2004) · Zbl 1058.76035
[3] Diehl, S., A conservation law with point source and discontinuous flux function, SIAM J. math. anal., 56, 388-419, (1996) · Zbl 0849.35142
[4] Gimse, T.; Risebro, N.H., Solution of Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. math. anal., 23, 635-648, (1992) · Zbl 0776.35034
[5] Holden, H.; Risebro, N.H., A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. math. anal., 26, 999-1017, (1995) · Zbl 0833.35089
[6] Isaacson, E.; Temple, B., Analysis of a singular hyperbolic system of conservation laws, J. diff. equations, 65, 250-268, (1986) · Zbl 0612.35085
[7] Jin, S.; Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. pure appl. math., 48, 235-277, (1995) · Zbl 0826.65078
[8] Karlsen, K.H.; Klingenberg, C.; Risebro, N.H., A relaxation scheme for conservation laws with a discontinuous coefficient, Math. comp., 73, 1235-1259, (2003) · Zbl 1078.65076
[9] Klar, A.; Pareschi, L.; Seaïd, M., Uniformly accurate schemes for relaxation approximations to fluid dynamic equations, Appl. math. lett., 16, 1123-1127, (2003) · Zbl 1040.76047
[10] Lattanzio, C.; Serre, D., Convergence of a relaxation scheme for hyperbolic systems of conservation laws, Numer. math., 88, 121-134, (2001) · Zbl 0983.35086
[11] J. LeVeque Randall, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich (1992). · Zbl 0847.65053
[12] Lighthill, M.J.; Whitham, J.B., On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. roy. soc. London, ser. A, 229, 317-345, (1955) · Zbl 0064.20906
[13] Natalini, R., Convergence to equilibrium for relaxation approximations of conservation laws, Comm. pure appl. math., 49, 795-823, (1996) · Zbl 0872.35064
[14] M. Seaïd, A. Klar, Asymptotic-preserving schemes for unsteady flow simulations, Comput. Fluids (in press).
[15] M. Seaïd, High-resolution relaxation scheme for the two-dimensional Riemann problems in gas dynamics, Numer. Meth. Partial Diff. Equat. (in press).
[16] Seaïd, M., Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions, Int. J. numer. meth. fluids, 46, 457-484, (2004) · Zbl 1060.76591
[17] Towers, J.D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. math. anal., 38, 681-698, (2000) · Zbl 0972.65060
[18] Van Leer, B., Towards the ultimate conservative difference schemes V. A second-order sequal to godunov’s method, J. comp. phys., 32, 101-136, (1979) · Zbl 1364.65223
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.