Energy-conserving Kansa methods for Hamiltonian wave equations

TL;DR

The paper addresses the challenge of numerically solving Hamiltonian wave equations while preserving energy. It develops a energy-conserving, meshfree discretization based on the least-squares Kansa method with kernel-based collocation and CN/CNAB time stepping, augmented by a quadratic energy constraint, and solves the resulting nonlinear constrained least-squares problem with a GSVD-Lagrange-Newton solver. The approach yields an energy-preserving scheme with competitive accuracy and notable computational efficiency, outperforming traditional secant-based solvers and showing strong long-time stability. This structure-preserving, flexible framework offers a practical path for reliable simulations of nonlinear wave phenomena and may extend to EC-constrained discretizations beyond Kansa.

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.

Figures (7)

• Figure 1: (Example 1 for PDE 1) The relative $L^2$ errors and relative energy errors vary over time for the EC-LS Kansa-CNAB method at different tolerances, ranging from $10^{-2}$ to $10^{-8}$, applied to solving the 2D linear wave equation.
• Figure 2: (Example 2 for PDE 2) Relative $L^2$ convergence profiles for PDE 2 solved using the EC-LS Kansa-CNAB method with Whittle-Matérn-Sobolev kernels of smoothness order $m = 3, 4, 5$, and $\epsilon = 5$, for $\tau = 5\times 10^{-4}$ and $T = 1$. Errors for ratios of oversampling $\gamma = (n_X+n_Y)/n_Z, \in \{1, 2, 3\}$ are shown individually, while errors for $\gamma \in \{4, 5, 6, 7\}$ are collectively represented in a shaded area.
• Figure 3: (Example 3 for PDEs 3-4) Comparison of energy conservation under different boundary conditions for the 2D Klein-Gordon equation: (a) Relative energy errors for PDE 3 under Dirichlet conditions; (b) Relative energy errors for PDE 4 under Neumann conditions.
• Figure 4: (Example 4 for PDE 2) Long-time numerical accuracy and energy conservation performance of different methods (LS Kansa-CNAB, MGAVF, and EC-LS Kansa-CNAB) at uniform and Halton point distributions. On (a) uniform points and (b) Halton points, the relative $L^2$ error (cyan, dark blue, and light blue corresponding to EC-LS Kansa-CNAB, MGAVF, and LS Kansa-CNAB) and the associated relative energy error (magenta, red, and light coral corresponding to EC-LS Kansa-CNAB, MGAVF, and LS Kansa-CNAB) are plotted respectively.
• Figure 5: (Example 5 for PDE 1) Numerical solutions and absolute error distribution of the 2D linear wave equation across various periods using the EC-LS Kansa-CN(AB) method. The left panel displays wave crests at the $10^{\mathrm{th}}$, $30^{\mathrm{th}}$, $50^{\mathrm{th}}$, and $70^{\mathrm{th}}$ wave periods, while the right panel shows wave troughs at the $20^{\mathrm{th}}$, $40^{\mathrm{th}}$, $60^{\mathrm{th}}$, and $80^{\mathrm{th}}$ wave periods.