We introduce a fast, constrained meshfree solver designed specifically to inherit energy conservation (EC) in second-order time-dependent Hamiltonian wave equations. For discretization, we adopt the Kansa method, also known as the kernel-based collocation method, combined with time-stepping. This approach ensures that the critical structural feature of energy conservation is maintained over time by embedding a quadratic constraint into the definition of the numerical solution. To address the computational challenges posed by the nonlinearity in the Hamiltonian wave equations and the EC constraint, we propose a fast iterative solver based on the Newton method with successive linearization. This novel solver significantly accelerates the computation, making the method highly effective for practical applications. Numerical comparisons with the traditional secant methods highlight the competitive performance of our scheme. These results demonstrate that our method not only conserves the energy but also offers a promising new direction for solving Hamiltonian wave equations more efficiently. While we focus on the Kansa method and corresponding convergence theories in this study, the proposed solver is based solely on linear algebra techniques and has the potential to be applied to EC constrained optimization problems arising from other PDE discretization methods.

翻译：本文提出一种快速、带约束的无网格求解器，专门设计用于在二阶时间依赖哈密顿波动方程中继承能量守恒特性。在离散化方面，我们采用Kansa方法（亦称基于核的配点法）并结合时间步进。该方法通过在数值解的定义中嵌入二次约束，确保能量守恒这一关键结构特征随时间得以保持。为应对哈密顿波动方程中的非线性特性及能量守恒约束带来的计算挑战，我们提出一种基于牛顿法并采用逐次线性化的快速迭代求解器。该新型求解器显著加速了计算过程，使本方法在实际应用中具有高效性。与传统割线法的数值比较突显了我们方案的竞争优势。结果表明，我们的方法不仅能保持能量守恒，还为更高效求解哈密顿波动方程提供了新的研究方向。尽管本研究聚焦于Kansa方法及相应收敛理论，但所提出的求解器完全基于线性代数技术，有望应用于其他偏微分方程离散化方法产生的能量守恒约束优化问题。