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

We propose a novel spectral method for the Allen--Cahn equation on spheres that does not necessarily require quadrature exactness assumptions. Instead of certain exactness degrees, we employ a restricted isometry relation based on the Marcinkiewicz--Zygmund system of quadrature rules to quantify the quadrature error of polynomial integrands. The new method imposes only some conditions on the polynomial degree of numerical solutions to derive the maximum principle and energy stability, and thus, differs substantially from existing methods in the literature that often rely on strict conditions on the time stepping size, Lipschitz property of the nonlinear term, or $L^{\infty}$ boundedness of numerical solutions. Moreover, the new method is suitable for long-time simulations because the time stepping size is independent of the diffusion coefficient in the equation. Inspired by the effective maximum principle recently 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. Further, we show that the new method is energy stable and equivalent to the Galerkin method if the quadrature rule exhibits sufficient exactness degrees. In addition, we propose an energy-stable mixed-quadrature scheme which works well even with randomly sampled initial condition data. We validate the theoretical results about the energy stability and the almost sharp maximum principle by numerical experiments on the 2-sphere $\mathbb{S}^2$.