Collocation discretizations of the transport equation with radial basis functions

In this paper, a new gridless method for numerically solving hyperbolic partial differential equations is presented. This method uses collocation based on Hermite interpolation at scattered sites at each time step. The basis can be chosen at each time step, which makes the approach adaptive and allows flexibility. The goal of this paper is to demonstrate the potential of this adaptive, flexible model which includes the ability to use scattered sites in a multi-dimensional setting with compactly supported radial functions. To illustrate the method and to compare it with known results, we analyze a certain transport problem in detail. When combined with an implicit time discretization and applied to smooth solutions, we show that the method yields spectral rates of convergence in the spatial domain. The implicit time discretization makes possible time steps of a size appropriate to the smoothness of the solution.