Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations

We introduce new Godunov-type semidiscrete central schemes for hyperbolic s ystems of conservation laws and Hamilton Jacobi equations. The schemes are based on the use of more precise information about the local speeds of prop agation and can be viewed as a generalization of the schemes from [A. Kurga nov and E. Tadmor, J. Comput. Phys. 160 (2000), pp. 241-282; A. Kurganov an d D. Levy, SIAM J. Sci. Comput. 22 (2000), pp. 1461-1488; A. Kurganov and G . Petrova A third-order semidiscrete genuinely multidimensional central sch eme for hyperbolic conservation laws and related problems Numer. Math., to appear] and [ A. Kurganov and E. Tadmor, J. Comput. Phys. 160 (2000), pp. 7 20-742]. The main advantages of the proposed central schemes are the high resolution , due to the smaller amount of the numerical dissipation, and the simplicit y. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they r espect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton Jacobi equations with convex and no nconvex Hamiltonians, and the incompressible Euler and Navier Stokes equati ons. The incompressibility condition in the latter equations allows us to t reat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.