\section{Introduction}\label{sec1:Intro}


There exists plenty of examples in the literature where stable approximations
of hyperbolic conservation laws exhibit
a linear error growth (or nearly linear growth) in time, 
even though stability of the numerical schemes should guarantee that the solution 
remains bounded, see
\cite{hesthaven2002nodal, cohen2006spatial, nordstrom2003high}.  
The reason behind this is the following:
the error equation for the time variation contains a forcing term generated by the
approximation or truncation errors and
this forcing term can trigger the unbounded growth of the error.\\
Simultaneously, there are other examples where the temporal error growth is
bounded \cite{abarbanel2000error, koley2009higher}.
Finally, in \cite{nordstrom2007error}, the author gives an explanation 
under what conditions the error is or is not bounded in time. 
The author works with SBP-SAT (Summation-by-Parts/Simultaneous-Approximation-Term) 
finite difference approximation and deduces
that the error behavior depends only on the choice of boundary condition of the problem. \\
If one considers waves in cavities or with periodic boundary conditions,
 linear growth is observed as it is investigated in \cite{hesthaven2002nodal},
where for inflow-outflow problems one obtains  boundedness.
In other words, if an appropriate boundary condition (sufficiently dissipative) 
is applied, the error is bounded. In this framework, the error behavior
does not depend on the internal 
discretization. 
In \cite{kopriva2017error}, the authors analyze the long-time behavior of the error
for discontinuous Galerkin spectral element methods (DGSEM).
They confirm the conclusion from \cite{nordstrom2007error} that the bounded error 
property is due to the dissipative boundary conditions, but in contrast
to \cite{nordstrom2007error}, in the DGSEM framework the internal approximation 
has indeed an influence on the behavior of the error. 
The choice of the numerical flux (upwind or  central) is essential for 
the magnitude of the error and the  speed at which the asymptotic error is 
reached. With the upwind flux one obtains better results.  \\
In this paper, we examine the long-time error behavior for a recent 
correction procedure via reconstruction (CPR) / flux reconstruction (FR) method.
The CPR/FR is a unifying  framework for several high-order methods such as discontinuous
Galerkin (DG), spectral difference (SD), spectral volume (SV), and
the residual distribution methods \cite{abgrall2018connection, huynh2014high},
and  includes through special choice of the nodal basis and of the correction matrix 
the DGSEM of \cite{kopriva2017error}. 
Here, we investigate not only the numerical flux, but also the selection of nodal 
basis (Gauß-Legendre and Gauß-Lobatto). We recognize that  the selection 
of the flux function is less  important than the choice of the nodal basis
for the error behavior. 
Using Gauß-Legendre basis in the approximation, the error is lower than in the Gauß-Lobatto
case  and the speed of attaining the error asymptotics  is even faster 
for most of the problems under consideration. 
The selection of the numerical flux 
has less influence  on the error behavior when using Gauß-Legendre nodes
than when applying a Gauß-Lobatto 
basis. Our investigation leads us to the conclusion that for many problems 
Gauß-Legendre points are the right choice, especially for low-order resolutions.
Our analysis extends and completes the investigation / predictions from 
\cite{kopriva2017error}
to a more general framework. \\ 
The paper is organized as follows:
in the second section, we repeat the main ideas of the SBP-CPR/FR methods 
and demonstrate the connection 
between CPR/FR and the DG framework. Then, in the section \ref{sec:model}
we present the model problem under consideration. In the next section
 \ref{sec:stability} we provide some approximation results and repeat the stability 
 analysis of the SBP-CPR/FR methods. 
Then, we extend the results from \cite{kopriva2017error}
to the linearly stable one-parameter family of Vincent \cite{vincent2011newclass}
and also consider in our investigation Gauß-Legendre nodes in the section \ref{sec:Error_equation}.
These nodes do not contain the boundary values 
in one element and this yields  a further error term in our error equation.
We focus on this additional error term and give an interpretation for it. 
We confirm our theoretical investigation by numerical tests in the section \ref{6_Numerical} which includes also one
example from \cite{kopriva2017error}
for comparison.
We mention some limitations of our results and finally, we summarize and
discuss these limitations. 
In the appendix, we show the relation between FR and DG and focus on stability conditions for FR methods
as described in \cite{vincent2011newclass}.
{"config": "arxiv", "file": "1806.01575/1_Introduction.tex"}