A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

We construct a new third-order semi-discrete genuinely multidimensional cen tral scheme for systems of conservation laws and related convection-diffusi on equations. This construction is based on a multidimensional extension of the idea, introduced in [17] - the use of more precise information about t he local speeds of propagation, and integration over nonuniform control vol umes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is i ndependent of O((1)/(Deltat)), allows us to pass to a limit as Deltat down arrow 0. This results in a particularly simple genuinely multidimensional s emi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstructi on. First, we introduce a less dissipative modification of the reconstructi on, proposed in [29]. Then, we generalize it for the computation of the two -dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes - simplicity, namely it does not employ Riemann solvers and characteristic de composition. This makes it a universal method, which can be easily implemen ted to a wide variety of problems. In this paper, the developed scheme is a pplied to the Euler equations of gas dynamics, a convection-diffusion equat ion with strongly degenerate diffusion, the incompressible Euler and Navier -Stokes equations. These numerical experiments demonstrate the desired accu racy and high resolution of our scheme.