A fully discrete approximation of the one-dimensional stochastic heat equation
This work provides a more efficient numerical method for solving a class of stochastic partial differential equations, improving upon existing error bounds and convergence results for researchers in computational stochastic analysis.
The paper develops a fully discrete approximation for the one-dimensional stochastic heat equation with multiplicative space-time white noise, using finite differences in space and a stochastic exponential method in time. The scheme avoids CFL restrictions and achieves improved error bounds in L^q(Ω) for globally Lipschitz coefficients, along with almost sure convergence, and provides conditions for convergence in probability for non-globally Lipschitz coefficients.
A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in $L^q(Ω)$, for all $q\geq2$, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.