A class of higher-order time-splitting Monte Carlo method for fractional Allen–Cahn equation

Abstract

In this paper, we introduce a novel class of higher-order time-splitting Monte Carlo method tailored for both fractional and classical Allen–Cahn equations. The proposed method integrates the spectral Monte Carlo method (SMC) with a time-splitting scheme, alternating between efficiently computing the linear propagator via the spectral Monte Carlo method and explicitly evaluating the nonlinear propagator. Numerical results for various $\alpha \in (0,2]$ demonstrate the method’s ability to achieve first-, second-, and fourth-order convergence rates, thereby confirming its effectiveness and accuracy.