Milstein Mittag-Leffler Integrator for Stochastic Semilinear Subdiffusion with Fractionally Integrated Multiplicative Noise

This paper investigates the strong convergence of a numerical method for stochastic semilinear subdiffusion equations characterized by a Caputo fractional derivative of order \(α ∈ (1/2,1)\), driven by fractionally integrated multiplicative noise. A spatio-temporal full discretization scheme, combining the Milstein Mittag-Leffler integrator and the finite element method, is developed and analyzed. The solution operator corresponding to the mild solution does not satisfy the semigroup property due to memory effects induced by the fractional derivative, complicating the proof of well-posedness. The presence of the nonlinear drift term and the high-order temporal discretization using the Milstein scheme make the strong convergence analysis more challenging compared to the case of stochastic linear subdiffusions. We establish the well-posedness of the underlying equation under certain assumptions, using Mainardi’s Wright function and an equivalent norm. The spatio-temporal strong convergence rate of the fully discrete solution is derived using fractional semigroup theory. The time convergence order obtained in this work is twice that of the Maruyama fractional exponential integrator presented by Noupelah et al. [Commun. Nonlinear Sci. 125 (2023)]. Numerical experiments are finally reported to validate the theoretical results.