Numerical solution of SPDEs on a class of two-dimensional domains and of random differential equations using PCE with exponential time differencing

The first part of this thesis deals with the numerical solution of semilinear parabolic stochastic partial differential equations on general two-dimensional C²-bounded domains. The existing exponential Euler time stepping scheme is used on two-dimensional domains, using a spectral approximation in space. Since the base functions are not given analytically, they are numerically approximated using a boundary element method combined with a contour integral method to solve nonlinear eigenvalue problems. An error analysis is given, and numerical experiments conclude the investigation of the method. The second part of the thesis compares the performance of non-intrusive and intrusive polynomial chaos expansion methods based on exponential time differencing schemes for a range of random ordinary and partial differential equations. It is shown in comprehensive numerical experiments that the two approaches are competitive for a range of different equations, but intrusive polynomial chaos becomes less efficient for polynomial nonlinearities of degree greater than two and breaks down for a reaction-diffusion equation exhibiting complex pattern formation behavior.