Numerical Methodologies for Solving Partial Differential Equations.

摘要

The numerical methods for solving systems of partial differential equations can be analyzed by decoupling the space and time discretizations and analyzing them independently. First a method is selected to discretize the differential equation in space and incorporate the boundary conditions. The spectrum of this discrete operator is then used as a guide to choose an appropriate method to integrate the equations through time. The dissipative effects of a numerical method are crucial to constructing reliable methods for conservation laws. This is particularly true when the solution is discontinuous as in a shock wave or contact discontinuity. Choosing an accurate method to accomplish each of these tasks, space and time discretization and incorporating artificial dissipation in the numerical solution, determines the success of the calculation. We will describe the methodologies used in each of these choices to construct reliable, accurate and efficient methods. 13 refs., 6 figs.