Better Accuracy for Simulations of Noisy Phenomena
Fast evolving physical phenomena can be described by mathematical equations containing noisy processes. Care is needed when numerically solving such equations, as naïve application of traditional methods can lead to large errors. A correction term, known as the Itô correction in the theory of stochastic differential equations, can be added to traditional methods to address the challenge. While the original Itô correction was designed to handle white noise, a recent study by a team of Pacific Northwest National Laboratory researchers proposed a generalization that can be applied to noisy processes of a wide range of characteristics.
The generalized Itô correction can be used to improve the quality of simulations containing noisy terms that state-of-the-art weather and climate models have started to include in their equations to help improve the simulation’s statistical properties and to facilitate the quantification of uncertainties. It is also applicable to a wide range of equations beyond weather and climate modeling.
Both fast-evolving and inherently random physical phenomena can appear noisy in numerical simulations. Numerical methods originally developed for deterministic and smooth phenomena can produce large errors when applied to noisy processes and can even lead to qualitatively different results. The concept of Itô correction, widely known as part of the theory of stochastic differential equations, can help address the challenge, but the classical Itô correction is only applicable to white noise. In this study, a generalized formulation of the Itô correction is derived for noise of any color, making it applicable to processes with memory and more suitable for many applications in weather, climate, and Earth system modeling. The generalized Itô correction is particularly helpful for the development of state-of-the-art weather and climate models, as noisy terms describing small-scale phenomena are being introduced to these models as part of the so-called stochastic parameterizations. The generalized Itô correction can help improve solution accuracy without requiring a complete redesign of the time-stepping methods in the original model codes. While this study was motivated by needs in atmosphere modeling, the formulation of the new Itô correction is general and applicable to a broad range of stochastic model equations.