Numerical solution of time-dependent stochastic partial differential equations using RBF partition of unity collocation method based on finite difference

Abstract

Meshfree methods based on radial basis functions (RBFs) are popular tools for the numerical solution of stochastic partial differential equations (SPDEs) due to their nice properties. However, the RBF collocation methods in global view have some disadvantages for the numerical solution of time-dependent SPDEs. Calculation of matrix condition number in the resulting dense linear systems indicates that the meshless method using global RBFs may be unstable at each realization to solve SPDEs. In order to avoid numerical instabilities in global RBF methods, we are interested in the use of RBF methods in local view for the numerical solution of time-dependent SPDEs. In this paper, the RBF partition of unity collocation method based on a finite difference scheme for the Gaussian random field (RBF-PU-FD) as a localized RBF approximation presented to deal with these issues. For this purpose, we simulate the Gaussian field with spatial covariance structure at a finite collection of predetermined collocation points. The matrices formed during the RBF-PU-FD method will be sparse and, hence, will not suffer from ill-conditioning and high computational cost. We will show that the method is viable through analyzing its numerical accuracy, CPU time, stability and sparsity structure. For the test problems, we perform 1000 realizations and statistical criterions such as mean, standard deviation, lower bound and upper bound of prediction are computed and evaluated using the Monte-Carlo method.