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 central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans.

As in the one-dimensional case, the small numerical dissipation, which is independent of \({\cal O}(\frac{1}{\Delta t})\), allows us to pass to a limit as \(\Delta t \downarrow 0\). This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, 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 decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme.

Authors and Affiliations

1. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA; e-mail: {kurganov,petrova}@math.lsa.umich.edu , , , , , , US

   Alexander Kurganov & Guergana Petrova

Received February 7, 2000 / Published online December 19, 2000

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