Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

Abstract. We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton–Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241–282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461–1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear and A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720–742. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide varietyof applications. At the same time, the developed schemes have an upwind nature, since theyrespect the directions of wave propagation bymeasuring the one-sided local speeds. This is whywe 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 nonconvex Hamiltonians, and the incompressible Euler and Navier–Stokes equations. The incompressibilitycondition in the latter equations allows us to treat them both in their conservative and transport form. We applyto these problems the central-upwind schemes, developed separatelyfor each of them, and compute the corresponding numerical solutions. Key words. multidimensional conservation laws and Hamilton–Jacobi equations, high-resolution semidiscrete central schemes, compressible and incompressible Euler equations