Further evaluating the generalized Itô correction for accelerating convergence of stochastic parameterizations with colored noise

Abstract. Stochastic parameterizations are increasingly used in numerical weather prediction to capture statistical properties of unresolved processes and model uncertainties. However, numerical methods developed for deterministic systems may fail to converge to physically meaningful solutions when applied to stochastic systems without modification. A recent study demonstrated the effectiveness of the generalized Itô correction in improving convergence and solution accuracy for a one-dimensional linear test problem with various noise spectra. In this work, we extend the analysis to two nonlinear systems: a modified one-dimensional Korteweg–de Vries equation and a two-dimensional nonlinear shear layer simulation relevant to numerical weather prediction. Both systems are subjected to stochastic advection with varying noise colors and magnitudes. We compare the convergence and solution accuracy of the Itô-corrected scheme to an uncorrected scheme, as well as its computational efficiency relative to a second-order Runge–Kutta method. Our results highlight the effectiveness of the generalized Itô correction in enhancing solution accuracy and convergence while maintaining computational efficiency.