In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geome...In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.展开更多
基金This research was supported by the National Natural Science Foundation of China 11271357,11271195 and 41504078by the CSC,the Foundation for Innovative Research Groups of the NNSFC 11321061 and the ITER-China Program 2014GB124005。
文摘In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.
