Abstract
In this paper we further develop the Method of Nearby Problems (MNP) for generating exact solutions to realistic partial differential equations by extending it to two dimensions. We provide an extensive discussion of the 2D spline fitting approach which provides C k continuity (continuity of the solution value and its first k derivatives) along spline boundaries and is readily extendable to higher dimensions. A detailed one-dimensional example is given to outline the general concepts, then the two-dimensional spline fitting approach is applied to two problems: heat conduction with a distributed source term and the viscous, incompressible flow in a lid-driven cavity with both a constant lid velocity and a regularized lid velocity (which removes the strong corner singularities). The spline fitting approach results in very small spline fitting errors for the heat conduction problem and the regularized driven cavity, whereas the fitting errors in the standard lid-driven cavity case are somewhat larger due to the singular behaviour of the pressure near the driven lid. The MNP approach is used successfully as a discretization error estimator for the driven cavity cases, outperforming Richardson extrapolation which requires two grid levels. However, MNP has difficulties with the simpler heat conduction case due to the discretization errors having the same magnitude as the spline fitting errors.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call