Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

Abstract

Nonlinear wave equations, such as Burgers equation and compound KdV–Burgers equation, are a class of partial differential equations (PDEs) with dissipation in Hamiltonian space, the numerical method of which plays an important role in complex fluid analysis. Based on the multi-symplectic idea, a new theoretical framework named generalized multi-symplectic integrator for a class of nonlinear wave PDEs with small damping is proposed in this paper. The generalized multi-symplectic formulation is introduced, and a twelve-point generalized multi-symplectic scheme, which satisfies two discrete modified conservation laws approximately as well as the local momentum conservation law accurately, is constructed to solve the first-order PDEs that derived from the compound KdV–Burgers equation. To test the generalized multi-symplectic scheme, several numerical experiments on the travelling front solution are carried out, the results of which imply that the generalized multi-symplectic scheme can simulate the travelling front solution accurately and satisfy the modified conservation laws well when step sizes and the damping parameter satisfy the inequality (41). It is more remarkable that the scheme (36) can be used to capture the shock wave structure of the compound KdV–Burgers equation within one data point, which can further illustrate the good structure-preserving property of the generalized multi-symplectic scheme (36). From the results of this paper, we can conclude that, similar to the multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property. For the existing of the excellent numerical properties, the generalized multi-symplectic method can be used to exposit some specific phenomena in the complex fluid.