Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications

TL;DR

This paper develops a time discretization method for stochastic geophysical fluid dynamics equations, proving convergence and existence of weak solutions, advancing numerical analysis in climate modeling.

Contribution

It introduces a novel convergence proof for an implicit Euler scheme applied to stochastic primitive equations with realistic boundary conditions.

Findings

Convergence of the Euler scheme for various geophysical equations

Existence of weak solutions in PDE and probabilistic sense

Applicable to oceanic, atmospheric, and coupled systems

Abstract

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and oceans, we study their time discretization by an implicit Euler scheme. From deterministic viewpoint the 3D Primitive Equations are studied with physically realistic boundary conditions. From probabilistic viewpoint we consider a wide class of nonlinear, state dependent, white noise forcings. The proof of convergence of the Euler scheme covers the equations for the oceans, atmosphere, coupled oceanic-atmospheric system and other geophysical equations. We obtain the existence of solutions weak in PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.