Phase error and stability of second order methods for hyperbolic problems. II

Abstract

Two new second order schemes are presented for the solution of hyperbolic systems in two space dimensions. The first method is a modification of the Lax-Wendreff method that dramatically reduces the numerical phase error compared with all other schemes that do not use data beyond a nine point lattice. The second scheme is a one parameter generalization of the rectangular form of Richtmyer's method. The scheme is shown to be stable (the stability proof is only formal since it is a nonlinear scheme) for all symmetric hyperbolic equations. The phase error and allowable time steps are functions of the free parameter. At one extreme the parameter can be chosen so as to yield maximal allowable time steps but with a phase error that is large compared with other second order methods. Alternately we can choose the free parameter so that the phase error is smaller than that of the Richtmyer type schemes but at the cost of a smaller permissible time step. Numerical experiments with the equations of dynamic elasticity are presented that confirm these conclusions.