On the implementation of multigrid methods for the numerical solution of partial differential equations

A number of experimental implementations of the multigrid algorithm for the solution of systems of partial differential equations have been produced. One program is applicable to simple nonlinear scalar equations, the others to linear equations, scalar and systems, which may be mildly stiff. All use nested grids and residual extrapolation techniques to compute solution and error estimates very economically. One version implements list based adaptive grids to further decrease both computation and storage needed for comparable problems. Each experiment was demonstrated using a set of problems with known solutions and the program performance or nonperformance discussed. Several techniques were examined to ensure that the system of difference equations representing a given problem would be convergent. The use of artificial viscosity was found to be practical in the general case, though for linear problems the use of one-sided differencing may be superior.