Preprint 1997-013

Non-oscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws

Abstract: We construct, analyze and implement a new non-oscillatory high-resolution scheme for two-dimensional Hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two-steps: starting with given cell averages, we first predict pointvalues which are based on non-oscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, non-oscillatory central scheme -- a natural extension of the one-dimensional second-order central scheme of Nessyahu \& Tadmor \cite{NT}.

As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the central scheme is 'genuinely multidimensional' in the sense that it does not necessitate dimensional splitting.

We prove that the scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the scheme is positive (in the sense of Lax \& Liu \cite{LL}). We demonstrate the application of our central scheme to several prototype two-dimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our central scheme solves with high resolution the intricate wave interactions in the so called double Mach reflection problem, \cite{WC}, without following the characteristics; and finally we report on the accurate ray solutions of a weakly Hyperbolic system, \cite{ER}, rays which otherwise are missed by dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high-resolution.

Paper:

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Title:

Non-oscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws

Author(s):

Guang-Shan Jiang, mailto:gsj@math.ucla.edu

Eitan Tadmor, mailto:tadmor@math.ucla.edu

Publishing information:

SIAM J. on Scientific Computating

Comments:

Submitted by:

<> April 28 1997.