Biological and Environmental Research - Earth and Environmental System Sciences
Earth and Environmental System Modeling

Improving Solution Accuracy and Convergence for Stochastic Physics Parameterizations with Colored Noise

TitleImproving Solution Accuracy and Convergence for Stochastic Physics Parameterizations with Colored Noise
Publication TypeJournal Article
Year of Publication2020
JournalMonthly Weather Review
Volume148
Number6
Pages2251-2263
Abstract / Summary

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. In previous studies, it has been demonstrated that a correction term, known in stochastic analysis as the Itô correction, can help improve solution accuracy for various deterministic numerical schemes and ensure convergence to the physically relevant solution without substantial computational overhead. The usual formulation of the Itô correction is valid only when the stochasticity is represented by white noise. In this study, a generalized formulation of the Itô 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 Itô 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 Itô correction allows for the use of significantly longer time steps and, hence, helps to reduce the computational cost of the numerical simulation.

URLhttp://dx.doi.org/10.1175/mwr-d-19-0178.1
DOI10.1175/mwr-d-19-0178.1
Journal: Monthly Weather Review
Year of Publication: 2020
Volume: 148
Number: 6
Pages: 2251-2263
Publication Date: 06/2020

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. In previous studies, it has been demonstrated that a correction term, known in stochastic analysis as the Itô correction, can help improve solution accuracy for various deterministic numerical schemes and ensure convergence to the physically relevant solution without substantial computational overhead. The usual formulation of the Itô correction is valid only when the stochasticity is represented by white noise. In this study, a generalized formulation of the Itô 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 Itô 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 Itô correction allows for the use of significantly longer time steps and, hence, helps to reduce the computational cost of the numerical simulation.

DOI: 10.1175/mwr-d-19-0178.1
Citation:
Stinis, P, H Lei, J Li, and H Wan.  2020.  "Improving Solution Accuracy and Convergence for Stochastic Physics Parameterizations with Colored Noise."  Monthly Weather Review 148(6): 2251-2263.  https://doi.org/10.1175/mwr-d-19-0178.1.