Time-stepping in Petrov-Galerkin methods based on cubic B-splines for compactons

Four numerical methods with first- to fourth-order of accuracy have been developed for the time integration of the Rosenau-Hyman K(2, 2) equation. The error in the solution and the invariants for the propagation of one-compacton, and the stability in collisions among compactons have been studied using these methods. Numerically-induced radiation has also been characterized by means of wavefront velocity and wavefront amplitude, showing that the self-similarity of the radiation wavepackets observed in the numerical results is a consequence of the time-stepping method. Among the four methods studied in this paper, the best results in terms of accuracy, computational cost, and stability have been obtained by means of using the second-order time integration method.