Adaptive Grids and Nonlinear Equations

Abstract

Since the equations of hydrodynamics are non-linear, a computational solution of these equations usually involves a numerical integration. This numerical integration is particularly difficult to perform in the presence of discontinuities of the solution called “shocks”. Although there are a number of devices available for obtaining the solution in the neighborhood of the shock, notably by von Neumann, Richtmeyer and Lax, these have the unfortunate property of altering the behavior of the solution in other regions. If conventional finite difference schemes are employed, instabilities arise in a number of cases.