An adaptive wavelet optimized finite difference B-spline polynomial chaos method for random partial differential equations

Abstract

In this paper, wavelet optimized finite difference B-spline polynomial chaos method is proposed for solving stochastic partial differential equations. The generalized polynomial chaos is applied by considering linear B-spline wavelet basis. Then, stochastic Galerkin projection is executed for evaluating the deterministic coefficients of the generalized polynomial chaos. In the next step, the system of equations is discretized by using Crank-–Nicolson scheme for time integration and for approximating the differential operators, central finite difference matrices are considered. An adaptive grid is generated using the linear B-spline generalized polynomial chaos for optimizing the numerical solution. The method is then tested on three problems namely heat equation with uncertain initial conditions, Burger’s equation with random initial conditions and Burger’s equation with random viscosity as well as uncertain initial conditions. For the three test problems, grid modifications are displayed by taking periodic boundary conditions into consideration. Mean and standard deviation are plotted for each test problem. Moreover, for the third test problem, computational time comparison is performed by computing CPU time taken by the proposed method and the CPU time taken by the finite difference method on a uniform grid.