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.

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

In this paper we consider the splitting method first introduced in rare event analysis. In this technique, the sample paths are split into R multiple copies at various stages to speed up the simulation. Given the cost, the optimization of the algorithm suggests taking all the transition probabilities to be equal; nevertheless, in practice, these quantities are unknown. We address this problem by presenting an algorithm in two phases.