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 $\mathbb{S}^2$.

翻译：我们提出了一种用于球面上Allen–Cahn方程的新型谱方法，该方法无需假设求积规则具有精确度。不同于依赖特定精确度阶数，我们基于Marcinkiewicz–Zygmund求积规则系统，采用受限等距关系来量化多项式被积函数的求积误差。新方法仅对数值解的多项式次数施加某些条件，即可导出最大值原理和能量稳定性，这与现有文献中通常依赖时间步长严格条件、非线性项的Lipschitz性质或数值解的$L^{\infty}$有界性的方法有本质区别。此外，由于时间步长与方程中的扩散系数无关，新方法适用于长时间模拟。受Li（Ann. Appl. Math., 37(2): 131–290, 2021）最近提出的有效最大值原理启发，我们发展了一种几乎精确的最大值原理，允许数值解在可控范围内偏离精确界。进一步地，我们证明了若求积规则具有足够的精确度阶数，新方法具有能量稳定性且等价于Galerkin方法。同时，我们提出了一种能量稳定的混合求积格式，该格式即使在随机采样的初始数据下也能良好运行。通过在二维球面$\mathbb{S}^2$上的数值实验，我们验证了关于能量稳定性和几乎精确最大值原理的理论结果。