A High-Order Meshless Linearly Implicit Energy-Preserving Method for Nonlinear Wave Equations on Riemannian Manifolds

In this paper, we propose a kernel-based meshless energy-preserving method for solving nonlinear wave equations on closed, compact, and smooth Riemannian manifolds. Our method employs the scalar auxiliary variable approach to transform the nonlinear term into a quadratic form, enabling a linearly implicit scheme that reduces computational time and has good energy conservation properties. Spatial discretization is achieved through a meshless Galerkin approximation in a finite-dimensional space spanned by Lagrange basis functions constructed from positive definite functions. The method demonstrates a high order of convergence without requiring an underlying mesh. Numerical experiments validate the theoretical analysis, confirming the convergence order and energy-preserving properties of the proposed method.