DMS-9805546 Xu-Dong Liu This project is concerned with several new numerical methods for solving multidimensional systems of conservation laws, including a new fully conservative scheme for multi-phase fluid calculations. The first contribution of this project is the extension of Friedrich's positivity principle from multi-dimensional symmetric linear systems to systems of conservation laws, which has become one of the guidelines for designing numerical methods. A family of positive schemes is constructed. Positive schemes are very robust, simple and of low cost. Many numerical experiments have shown that positive schemes are among the best 2nd order accurate high resolution methods. This is also the first work of this type which contains theoretical results for scheme design in multi-dimensional hyperbolic systems. The second contribution of this project is the new Convex Essential-Non-Oscillatory (CENO) schemes for multi-dimensional hyperbolic systems. The scheme can be implemented in component-wise fashion. Therefore its apparent advantages are: (1) No complete set of eigenvectors is needed and hence weakly hyperbolic systems can be solved. (2) Component-wise limiting is twice as fast as field-by-field limiting in each space dimension, which makes Convex ENO one of the fastest existing schemes. (3) The component-wise version of the scheme is simple to program. In addition, (4) the Convex ENO scheme is very robust. The third contribution of this project is the introduction of a fully conservative method for multi-phase flow problems. This new idea enables us to avoid the spurious oscillations near material interfaces common to all other conservative schemes. This is done through the addition of a general equation of state. The new scheme works essentially for mixture of any fluids such as gamma-law gas, water and JWL (explosive material). The new idea works in any space dimension and is scheme-independent, which means it should apply to a typical users' existi ng code. Preliminary numerical experiments show that this scheme is very promising. This project is aimed at solving real world problems and is intended to have a significant impact on semiconductor device modeling, underwater and solid explosives modeling, computational fluid dynamics, magneto-hydrodynamics, and many other applications, which are all a part of high-performance computing. The principal goals are: (1) to design and improve numerical methods for more efficiency, simplicity, and robustness; (2) to improve computer simulation of multi-phase fluids. The methods reported in this project are a step towards this goal.
