An RBF-LOD Method for Solving Stochastic Diffusion Equations

In this study, we introduce an innovative approach to solving stochastic equations in two and three dimensions, leveraging a time-splitting strategy. Our method combines radial basis function (RBF) spatial discretization with the Crank–Nicolson scheme and the local one-dimensional (LOD) method for temporal approximation. To navigate the probabilistic space inherent in these equations, we employ the Monte Carlo method, providing accurate estimates for expectations and variations. We apply our approach to tackle challenging problems, including two-dimensional convection-diffusion and Burgers’ equations, resulting in reduced computational and memory requirements. Through rigorous testing against diverse problem sets, our methodology demonstrates efficiency and reliability, underscoring its potential as a valuable tool in solving complex multidimensional stochastic equations. We have validated the method’s stability and showcased its convergence as the number of collocation points increases. These findings serve as compelling evidence of the suggested method’s convergence properties.