Numerical methods in hydrodynamic calculations

The elements of the theory of finite difference methods for the numerical solution of systems of partial differential equations, and in particular the equations of compressible fluid dynamics, are presented. The notions of numerical stability, consistency, order of accuracy, etc., of finite difference schemes are defined, and examples of each notion are given using both the linear heat equation and the linear wave equation. The integral or weak formulation of the hydrodynamic equations is derived, and from this the Rankine-Hugoniot jump conditions for shocks and the conditions that hold at contact discontinuities are formulated. The general theory of the viscosity method of von Neumann and Richtmyer for following shocks is derived, and the most common difference methods for the Lagrangian and the Eulerian representations are presented. A brief historical review of LLL's two-dimensional hydrodynamics codes is presented.