Finite Difference Scheme for Calculating Problems in Two Space Dimensions and Time

摘要

In 1950 von Neumann and Richtmyer proposed the artificial viscosity method for calculating problems in hydrodynamics. The technique was described for one-dimensional flow with the Lagrange form of the hydrodynamic differential equations. The region of flow was divided into a finite mesh of grid points at which the various parameters could be specified. The differential equations were then approximated by a second-order-difference scheme involving these grid-point parameters. The approach taken at LRL was to develop a second-order-difference scheme that used the minimum number of grid points and hence had the minimum implicit artificial diffusion. The grid is then stabilized by adding an artificial viscosity to the equations of motions. In this manner the operation of the artificial viscosity is known, and its magnitude can be compared with the magnitude of physical stresses in the problem. The paper demonstrates the effectiveness of the program for solving a range of physical problems. Examples are given in elasticity, elastic-plastic flow, seismology and gas dynamics.