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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. 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 variety of applications.At the same time, the developed schemes have an upwind nature, since they respect 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 nonconvex Hamiltonians, and the incompressible Euler and Navier-Stokes equations. The incompressibility condition in the latter equations allows us to treat 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.

Abstract. A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002]. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously.Here, we introduce an improved second-order central-upwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one-and two-dimensional examples.

We report here on our numerical study of the two-dimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1-D configurations, the 2-D case consists of a plethora of geometric wave patterns that pose a computational challenge for high-resolution methods. The main feature in the present computations of these 2-D waves is the use of the Riemann-solvers-free central schemes presented by Kurganov et al. This family of central schemes avoids the intricate and timeconsuming computation of the eigensystem of the problem and hence offers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high resolution, the various features observed in the earlier, more expensive computations.

Abstract. We present one-and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.Mathematics Subject Classification. 65M06, 35L65.