Xu-Dong Liu

Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes

The positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in [X.-D. Liu and P. D. Lax, J. Fluid Dynam., 5 (1996), pp. 133--156]. Some numerical experiments presented there show how well the method works. In this paper we use positive schemes to solve Riemann problems for two-dimensional gas dynamics.

Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I

This is the first paper in a series in which a class of nonoscillatory high order accurate self-similar local maximum principle satisfying (in scalar conservation law) shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed. In this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented, using scalar one-dimensional initial value problems as a model. For this model, the schemes are made to satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modifications, imposed as needed. The first enforces a local maximum principle; the second guarantees that no new extrema develop. The schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [Math. Comp., 49 (1987), pp. 105–121, Math. Comp...

Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II

Kurganov and Tadmor have developed a numerical scheme for solving the initial value problem for hyperbolic systems of conservation laws. They showed that in the scalar case their scheme satisfies a local maximum-minimum principle i.e., the solution at future is bounded above and below by the solution at current locally. In this paper we show that this scheme is positive in the sense of Friedrichs for systems as well. We present the scheme of Kurganov and Tadmor as a convex combination of composites of positive schemes. Since each component of a composite scheme is bounded in the l2 norm, so is the convex combination of the composites. To achieve second order accuracy in time, we use a Runge-Kutta type scheme due to Shu and Osher. We present two numerical experiments to add to the ones carried out by Kurganov and Tadmor.

A General Technique for Eliminating Spurious Oscillations in Conservative Schemes for Multiphase and Multispecies Euler Equations

Standard conservative discretizations of the compressible Euler equations have been shown to admit nonphysical oscillations near some material interfaces. For example, the calorically perfect Euler equations admit these oscillations when both temperature and gamma jump across an interface, but not when either temperature or gamma happen to be constant. These nonphysical oscillations can be alleviated to some degree with a nonconservative modification of the total energy computed by solving a coupled evolution equation for the pressure. In this paper, we develop and illustrate this method for the thermally perfect Euler equations.

Convex ENO Schemes for Hamilton-Jacobi Equations

In one dimension, viscosity solutions of Hamilton–Jacobi (HJ) equations can be thought as primitives of entropy solutions for conservation laws. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations even in several dimensions. In this paper, we construct convex ENO (CENO) schemes for HJ equations. This construction is a generalization from the work by Liu and Osher on CENO schemes for conservation laws. Several numerical experiments are performed. L1 and L∞ error and convergence rate are calculated as well.