Date of Award
Doctor of Philosophy (PhD)
Traditional finite difference methods for solving the partial differential equations (PDEs) associated with wave and heat transport often perform poorly when used in domains that feature jump discontinuities in model parameter values (interfaces). We present a radial basis function-derived finite difference (RBF-FD) approach that solves these types of problems to a high order of accuracy, even when curved interfaces and variable model parameters are present.
The method generalizes easily to a variety of different problem types, and requires only the inversion of small, well-conditioned matrices to determine stencil weights that are applied directly to data that crosses an interface. These weights contain all necessary information about the interface (its curvature; the contrast in model parameters from one side to the other; variability of model parameter value on either side), and no further consideration of the interface is necessary during time integration of the numerical solution.
Martin, Bradley Pifer, "Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces" (2016). Applied Mathematics Graduate Theses & Dissertations. 79.