A meshless structure-preserving quasi-interpolation method for solving Allen–Cahn equations on spheres

This paper proposes a novel meshless structure-preserving scheme for solving Allen–Cahn equations on spheres. Within the scalar auxiliary variable (SAV) framework, we introduce a kernel-based spherical quasi-interpolation method for spatial discretization on scattered collocation points, complemented by appropriate time integration schemes. Our approach stands out by ensuring unconditional energy stability and overcoming the limitations of traditional mesh-based techniques. Quasi-interpolation methods are known for their simplicity and adaptability to various kernel properties, eliminating the need for large linear systems and making the scheme easy to implement. We provide rigorous theoretical analysis to validate the proposed spherical quasi-interpolation SAV scheme, including well-posedness, energy stability, and error estimates. Numerical examples further demonstrate the convergence of the quasi-interpolation method and the accuracy of simulating Allen–Cahn equations on the sphere.