  For practical applications, the long time behaviour of the error of numerical
solutions to time-dependent partial differential equations is very important.
Here, we investigate this topic in the context of hyperbolic conservation laws
and flux reconstruction schemes, focusing on the schemes in the discontinuous
Galerkin spectral element framework. For linear problems with constant
coefficients, it is well-known in the literature that the choice of the
numerical flux (e.g. central or upwind) and the selection of the polynomial
basis (e.g. Gau{\ss}-Legendre or Gau{\ss}-Lobatto-Legendre) affects both the
growth rate and the asymptotic value of the error. Here, we extend these
investigations of the long time error to variable coefficients using both
Gau{\ss}-Lobatto-Legendre and Gau{\ss}-Legendre nodes as well as several
numerical fluxes. We derive conditions guaranteeing that the errors are still
bounded in time. Furthermore, we analyse the error behaviour under these
conditions and demonstrate in several numerical tests similarities to the case
of constant coefficients. However, if these conditions are violated, the error
shows a completely different behaviour. Indeed, by applying central numerical
fluxes, the error increases without upper bound while upwind numerical fluxes
can still result in uniformly bounded numerical errors. An explanation for this
phenomenon is given, confirming our analytical investigations.
