The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
A conservative spectral method for several two-dimensional nonlinear wave equations
Journal of Computational Physics
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
Journal of Computational Physics
A Local Discontinuous Galerkin Method for KdV Type Equations
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Multisymplectic box schemes and the Korteweg-de Vries equation
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Local discontinuous Galerkin methods for nonlinear dispersive equations
Journal of Computational Physics
Local discontinuous Galerkin methods for nonlinear Schrödinger equations
Journal of Computational Physics
Journal of Computational Physics
Multi-symplectic integration of the Camassa-Holm equation
Journal of Computational Physics
On the multisymplecticity of partitioned Runge-Kutta and splitting methods
International Journal of Computer Mathematics - Splitting Methods for Differential Equations
Multisymplecticity and wave action conservation
Mathematics and Computers in Simulation
Splitting multisymplectic integrators for Maxwell's equations
Journal of Computational Physics
Applied Numerical Mathematics
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In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov-Kuznetsov (ZK) equation and the Kadomtsev-Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.