The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.

翻译：本文采用空间谱方法与时间上的前向和后向Euler-Maruyama格式，对球面上由加性各向同性Wiener噪声驱动的随机热方程进行近似。谱近似基于球谐函数级数展开的截断。针对初始条件与驱动噪声的给定正则性，推导了Euler-Maruyama方法的最优强收敛速率。除强收敛外，还证明了期望与二阶矩的收敛性，其中二阶矩近似以两倍强收敛速率收敛。数值模拟验证了理论结果。