A Meshless Symplectic Algorithm for multi-variate Hamiltonian PDEs with Radial Basis Approximation

Based on radial basis approximation, in this paper we propose two methods to discretize the problem of multi-variate Hamiltonian system. One is discretizing the system and finding out the corresponding discrete Hamiltonian functional, which will be conserved with respect to the time. The other is discretizing the Hamiltonian functional and deriving the corresponding discrete Hamiltonian system. This helps open a new area of research in developing the expected meshless symplectic algorithm for multi-variate Hamiltonian systems with the scattered data points. Theoretical estimates including the truncation error and the global error are given. Numerical experiments verify the theoretical results.As numerical experiments show, the schemes are easy to implement with the scattered knots. Furthermore, the schemes possesss a long-time tracking capability for these Hamiltonian systems.