SEMI-IMPLICIT MILSTEIN SCHEME FOR STOCHASTIC GINZBURG-LANDAU EQUATIONS
DOI:
https://doi.org/10.18173/2354-1059.2026-0028Keywords:
Ginzburg-Landau equations; Milstein scheme; Simulation; Stochastic differential equationAbstract
We propose a semi-implicit Milstein scheme for a class of stochastic differential equations with super-linearly growing dissipative drift coefficients, in particular on the stochastic Ginzburg–Landau equation. The nonlinear cubic drift is treated implicitly, while the diffusion and Milstein correction terms are evaluated explicitly. This construction preserves the dissipative structure of the underlying equation and avoids the use of taming or truncation techniques. Under the assumption that the diffusion coefficient possesses bounded derivatives up to second order, we establish the well-posedness of the proposed numerical scheme and derive uniform moment bounds of arbitrary order. We further prove that the proposed approximation converges strongly to the exact solution with order one in L2. Numerical experiments confirm the theoretical convergence rate and demonstrate the stability of the method for large initial data.
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