Convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation
Kavin Rajasekaran, Niklas Sapountzoglou
TL;DR
This work analyzes a stochastic non-linear parabolic equation with multiplicative noise and Neumann boundary conditions, proposing a TPFA finite-volume scheme in space and a semi-implicit Euler method in time. Under mild regularity assumptions on the initial data and diffusion terms, it establishes stability and regularity results for the exact and time-discrete solutions. The main contribution is a rigorous L2-space-time convergence rate for the fully discrete scheme, with an error bound of order Υ(τ + h^2 + h^2/τ). The results extend existing convergence-rate theory to nonzero convection-divergence terms and provide a practical framework for numerical simulations of stochastic diffusion-adiabatic-type systems.
Abstract
In this contribution, we provide convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions. More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization by a semi-implicit Euler scheme with respect to time and a two-point flux approximation finite volume scheme with respect to space and the variational solution. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.