Fourth order difference methods for the initial boundary-value problem for hyperbolic equations

Abstract

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation u t = − c u x {u_t} = - c{u_x} . A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.