Energy-conserving Kansa methods for Hamiltonian wave equations

TL;DR

This paper presents a novel energy-conserving meshfree solver for Hamiltonian wave equations that combines the Kansa method with a fast iterative Newton-based solver to ensure energy conservation and computational efficiency.

Contribution

The paper introduces a new energy-conserving Kansa method with a Newton-based solver for Hamiltonian wave equations, enhancing efficiency and structural preservation.

Findings

The method effectively conserves energy over time.

Numerical results show competitive performance compared to traditional methods.

The solver accelerates computations significantly for practical applications.

Abstract

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.