This study presents a novel meshless symplectic and multi-symplectic local radial basis function (RBF) collocation method (LRBFCM) for Hamiltonian partial differential equations. Specifically, the nonlinear wave equation and nonlinear Schrödinger equation are considered. The discretization in space is based on LRBFCM and then in time by symplectic integrator. The conservativeness of the method is explored and the accuracy is assessed. The LRBFCM is simple and efficient, since it can avoid the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. Numerical experiments with uniform knots and random knots are designed to illustrate the effectiveness of the method.

本研究提出了一种用于哈密顿偏微分方程的新型无网格辛和多辛局部径向基函数 (RBF) 搭配方法 (LRBFCM)。具体地，考虑非线性波动方程和非线性薛定谔方程。空间离散化基于LRBFCM，然后通过辛积分器在时间上离散化。探索该方法的保守性并评估其准确性。LRBFCM 简单高效，因为它可以避免全局 RBF 方法的病态问题和形状参数敏感性。设计了均匀结和随机结的数值实验来说明该方法的有效性。