A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems |
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Authors: | Alexander Kurganov Guergana Petrova |
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Institution: | (1) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA; e-mail: {kurganov,petrova}@math.lsa.umich.edu , US |
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Abstract: | Summary. 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 , allows us to pass to a limit as . 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.
Received February 7, 2000 / Published online December 19, 2000 |
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Keywords: | Mathematics Subject Classification (1991): 65M10 65M05 |
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