  Flux reconstruction provides a framework for solving partial differential
equations in which functions are discontinuously approximated within elements.
Typically, this is done by using polynomials. Here, the use of radial basis
functions as a methods for underlying functional approximation is explored in
one dimension, using both analytical and numerical methods. At some mesh
densities, RBF flux reconstruction is found to outperform polynomial flux
reconstruction, and this range of mesh densities becomes finer as the width of
the RBF interpolator is increased. A method which avoids the poor conditioning
of flat RBFs is used to test a wide range of basis shapes, and at very small
values, the polynomial behaviour is recovered. Changing the location of the
solution points is found to have an effect similar to that in polynomial FR,
with the Gauss--Legendre points being the most effective. Altering the location
of the functional centres is found to have only a very small effect on
performance. Similar behaviours are determined for the non-linear Burgers'
equation.
