Mathematical Researches

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Meshfree methods based on radial basis functions (RBFs) are popular tools for the numerical solution of parabolic stochastic partial differential equations (PSPDEs). However, the RBF collocation methods in the global view have some disadvantages for the numerical solution of PSPDEs. Condition number in the resulting dense linear systems indicates that the meshless method using global RBFs may be unstable at each realization to solve PSPDEs. In order to avoid numerical instabilities in the global view, we are interested in the use of RBF methods in the local view for the numerical solution of PSPDEs. 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 a spatial covariance structure at a finite collection of 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. For the test problems, we perform 1000 realizations and statistical criterions such as mean, standard deviation, lower bound and upper bound of prediction are evaluated using the Monte-Carlo method.
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Keywords: stochastic partial differential equations, meshless methods, RBF collocation, partition of unity method.

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