Semigroup Splitting And Cubature Approximations For The Stochastic Navier-Stokes Equations
This work provides theoretical convergence guarantees for high-order probabilistic numerical methods in a challenging infinite-dimensional setting, relevant to researchers in stochastic PDEs and computational fluid dynamics.
The authors develop high-order numerical methods for approximating the marginal distribution of solutions to the stochastic Navier-Stokes equations on a 2D torus, proving convergence rates for a splitting scheme and cubature on Wiener space applied to a spectral Galerkin discretization. Numerical simulations confirm the methods' applicability.
Approximation of the marginal distribution of the solution of the stochastic Navier-Stokes equations on the two-dimensional torus by high order numerical methods is considered. The corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on Wiener space applied to a spectral Galerkin discretisation of degree $N$. While the estimates exhibit a strong $N$ dependence, convergence is obtained for appropriately chosen time step sizes. Results of numerical simulations are provided, and confirm the applicability of the methods.