Research On Numerical Methods For Nonlinear Hyperbolic Partial Differential Equations

Many fluid mechanics problems such as aviation, scene, and oil recovery, end in nonlinear hyperbolic partial differential equations (also call conservation laws) problems. The basic difficulty with those problems is that the solutions develop discontinuities. In this paper, chapter I is devoted to mathematical theory which include weekly solution, entropy condition, physical solution, and some fundamental theorem. The reason for discontinuities arising in solution of hyperbolic equations and why high-resolution explicit schemes generate oscillations near discontinuities, is theoretically analyzed. Some current methods are given in chapter II. These include explicit artificial viscosity, implicit artificial viscosity, and self-adjusting hybrid schemes. In last chapter, we consider a new technique-"one of two'' method that can smooth oscillations near discontinuities, and compare our results with those of the related methods.