B-spline collocation for two dimensional, time-dependent, parabolic PDEs

Abstract

In this thesis, we consider B-spline collocation algorithms for solving two-dimensional in space, time-dependent parabolic partial differential equations (PDEs), defined over a rectangular region. We propose two ways to solve the problem: (i) The Method of Surfaces: Discretizing the problem in one of the spatial domains, we obtain a system of one-dimensional parabolic PDEs, which is then solved using a one-dimensional PDE system solver. (ii) Two-dimensional B-spline collocation: The numerical solution is represented as a bi-variate piecewise polynomial with unknown time-dependent coefficients. These coefficients are determined by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain, i.e., we collocate simultaneously in both spatial dimensions. This leads to an approximation of the PDE by a large system of time-dependent differential algebraic equations (DAEs), which we then solve using a high quality DAE solver.