An adaptive stochastic investigation of partial differential equations using wavelet collocation generalized polynomial chaos method

Starting from offering good solution to differential equations to capturing the nonlinear-ity, wavelets have gained more and more attention even for handling the uncertainties present in a physical system. In this paper, wavelet collocation approach has been united with the polynomial chaos expansion for numerically solving the stochastic partial differential equations. This approach is associated with the concept of autocorrelation functions of compactly supported Daubechies scaling functions. First of all, we make use of linear B-spline basis in generalized polynomial chaos. Then, in order to separate the randomness, Galerkin projection is executed on uncertain data and solution variables. After that, connection coefficients are calculated using Daubechies wavelets for approx-imating the differential operators. Moreover, fast algorithms are known to speed up the numerical scheme, so, we have executed a class of fast algorithms, on the basis of fast wavelet transform. Also, in order to handle the condition number, we have used a good diagonal preconditioning technique which makes the condition number of the matrices bounded. We have also executed an adaptive scheme which will focus on capturing the essential features of the solution by modifying the grids as well as reducing the CPU time. The scheme has been tested along with adaptivity on three stochastic partial differential equations with uncertain initial data and the results obtained from the proposed method are quite promising.(c) 2023 Elsevier B.V. All rights reserved.