Stable numerical methods for conservation laws with discontinuous flux function.

*(English)*Zbl 1088.65080Summary: 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 |

##### Keywords:

Conservation laws with discontinuous flux function; Relaxation approximation; Characteristics method; Traffic flow models; Euler method; Lagrange method; Stability; Numerical results; Two-phase flows
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\textit{M. Seaïd}, Appl. Math. Comput. 175, No. 1, 383--400 (2006; Zbl 1088.65080)

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