Geometric Numerical Integration for Peakon b-Family Equations (Communications in Computational Physics, 2018)
In this paper, we study the Camassa-Holm equation and the Degasperis-Procesiequation. The two equations are in the family of integrable peakon equations,and both have very rich geometric properties. Based on these geometric structures, weconstruct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many commonproperties, however, they also have the significant differences, for example, thereexist the shock wave solutions for the Degasperis-Procesi equation. By using the symplecticFourier pseudo-spectral integrator, we simulate the peakon solutions of the twoequations. To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods. In thenumerical experiments, comparisons of these two kinds of methods are presented interms of accuracy, computational cost and invariants preservation.