Iterated $\theta$-method for hyperbolic equations

The iterated $\theta$-methods employing residue smoothing for finding both steady state and time-accurate solutions of semidiscrete hyperbolic differential equations are analysed. By the technique of residue smoothing the stability condition is considerably relaxed, so that larger time steps are allowed which improves the efficiency of the method. The additional computational effort involved by the explicit smoothing technique used here is rather low when compared with its stabilizing effect. However, in the case where time-accurate solutions are desired, the overall accuracy may be decreased. This paper investigates the effect of residue smoothing on both the stability and accuracy, and presents a number of explicitly given methods based on the iterated implicit midpoint rule ($\theta$ = 1/2). Numerical examples confirm the theoretical results.