  Stochastic parameterizations are used in numerical weather prediction and
climate modeling to help capture the uncertainty in the simulations and improve
their statistical properties. Convergence issues can arise when time
integration methods originally developed for deterministic differential
equations are applied naively to stochastic problems. (Hodyss et al 2013, 2014)
demonstrated that a correction term to various deterministic numerical schemes,
known in stochastic analysis as the It\^o correction, can help improve solution
accuracy and ensure convergence to the physically relevant solution without
substantial computational overhead. The usual formulation of the It\^o
correction is valid only when the stochasticity is represented by {\it white}
noise. In this study, a generalized formulation of the It\^o correction is
derived for noises of any color. The formulation is applied to a test problem
described by an advection-diffusion equation forced with a spectrum of fast
processes. We present numerical results for cases with both constant and
spatially varying advection velocities to show that, for the same time step
sizes, the introduction of the generalized It\^o correction helps to
substantially reduce time integration error and significantly improve the
convergence rate of the numerical solutions when the forcing term in the
governing equation is rough (fast varying); alternatively, for the same target
accuracy, the generalized It\^o correction allows for the use of significantly
longer time steps and hence helps to reduce the computational cost of the
numerical simulation.
