In this thesis, the numerical approximation of statistics of solutions to partial differential equations with random data is studied. There are many numerical methods for solving models under uncertainty. In this thesis, the stochastic collocation method with sparse grids is the one that is used. The main aim is to compare three different types of collocation techniques for solving linear and nonlinear models. Moreover, we investigate the effect of using alternative methods for solving the sample problem that is obtained from using the collocation method. For both linear and nonlinear models, it is shown that the Gauss-Patterson sparse grid provides the most accurate approximation to solution statistics compared to the Clenshaw-Curtis and Chebyshev-Gauss-Lobatto grids. However, the efficiency of the Gauss-Patterson grid is not clear throughout our numerical experiments in the case of having a shock wave or an oscillation in the solution. It is also found that the efficiency of the collocation method is related to the efficiency of the underlying method which is used for solving the sample problem. Specifically, the first-order forward time central space scheme is more efficient than the backward time central space scheme for solving the convection-diffusion equation with a step function as initial data. Moreover, if the Peclet number does not satisfy the stability restriction then the streamline diffusion is more efficient than the Galerkin approximation. In the case of a spatially dependent random field, it is shown that the three type of collocation techniques have almost the same error when an alternative norm is used for estimating the error in the solution. However, when the second norm is used for estimating the error in solution statistics (mean and variance) the superiority of Gauss-Patterson grid is clear.
|Date of Award||1 Aug 2019|
- The University of Manchester
|Supervisor||David Silvester (Supervisor) & Catherine Powell (Supervisor)|