Breaking quadrature exactness: A spectral method for the Allen--Cahn equation on spheres

We present a novel spectral method for the Allen-Cahn equation on spheres, eliminating the reliance on conventional quadrature exactness conditions. By replacing these conditions with a restricted isometry relation derived from Marcinkiewicz--Zygmund quadrature systems, our method achieves precise control over quadrature errors for polynomial integrands. This theoretical advancement enables the use of substantially more choices of quadrature points than classical spectral methods while maintaining rigorous error bounds. The proposed method requires only mild constraints on the polynomial degree of numerical solutions to establish both the maximum principle and energy stability, representing a considerable departure from existing techniques that depend on restrictive time stepping sizes, Lipschitz property of the nonlinear term, or $L^{\infty}$ boundedness of numerical solutions. Notably, our method permits time stepping sizes independent of the diffusion coefficient, making it suitable for long-time simulations. Inspired by the effective maximum principle proposed by Li (Ann. Appl. Math., 37(2): 131--290, 2021), we develop an almost sharp maximum principle that allows controllable deviation of numerical solutions from the sharp bound with large time stepping sizes. Furthermore, we prove that when the quadrature rule attains sufficient exactness, our method preserves energy stability and coincides mathematically with the Galerkin method. In addition, we propose an energy-stable mixed-quadrature scheme which works well even with randomly sampled initial condition data. Our numerical experiments on $\mathbb{S}^2$ validate the theoretical results about the energy stability and the almost sharp maximum principle.