Analysis of smoothing operators in the solution of partial differential equations by explicit difference schemes

A smoothing technique for the “preconditioning” of the right-hand side of semidiscrete partial differential equations is analyzed. For a parabolic and a hyperbolic model problem, optimal smoothing matrices are constructed which result in a substantial amplification of the maximal stable integration step of arbitrary explicit time integrators when applied to the smoothed problem. This smoothing procedure is illustrated by integrating both linear and nonlinear parabolic and hyperbolic problems. The results show that the stability behaviour is comparable with that of the Crank-Nicolson method; furthermore, if the problem belongs to the problem class in which the time derivative of the solution is a smooth function of the space variables, then the accuracy is also comparable with that of the Crank-Nicolson method.