A numerical method in which the Rankine-Hugoniot condition is enforced at the discrete level is developed. The simple format of central discretization in a finite volume method is used together with the jump condition to develop a simple and yet accurate numerical method free of Riemann solvers and complicated flux splittings. The steady discontinuities are captured accurately by this numerical method. The basic idea is to fix the coefficient of numerical dissipation based on the Rankine-Hugoniot (jump) condition. Several numerical examples for scalar and vector hyperbolic conservation laws representing the inviscid Burgers equation, the Euler equations of gas dynamics, shallow water equations and ideal MHD equations in one and two dimensions are presented which demonstrate the efficiency and accuracy of this numerical method in capturing the flow features.

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The relation between the Lattice Boltzmann Method, which has recently become popular, and the Kinetic Schemes, which are routinely used in Computational Fluid Dynamics, is explored. A new discrete velocity model for the numerical solution of Navier{Stokes equations for incompressible uid ow is presented by combining boththe approaches. The new scheme can beinterpreted as a pseudo-compressibility method and, for a particular choice of parameters, this interpretation carries over to the Lattice Boltzmann Method.

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