  High-order finite volume and finite element methods offer impressive accuracy
and cost efficiency when solving hyperbolic conservation laws with smooth
solutions. However, if the solution contains discontinuities, these high-order
methods can introduce unphysical oscillations and severe
overshoots/undershoots. Slope limiters are an effective remedy, combating these
oscillations by preserving monotonicity. Some limiters can even maintain a
strict maximum principle in the numerical solution. They can be classified into
one of two categories: \textit{a priori} and \textit{a posteriori} limiters.
The former revises the high-order solution based only on data at the current
time $t^n$, while the latter involves computing a candidate solution at
$t^{n+1}$ and iteratively recomputing it until some conditions are satisfied.
These two limiting paradigms are available for both finite volume and finite
element methods.
  In this work, we develop a methodology to compare \textit{a priori} and
\textit{a posteriori} limiters for finite volume solvers at arbitrarily high
order. We select the maximum principle preserving scheme presented in
\cite{zhang2011maximum, zhang2010maximum} as our \textit{a priori} limited
scheme. For \textit{a posteriori} limiting, we adopt the methodology presented
in \cite{clain2011high} and search for so-called \textit{troubled cells} in the
candidate solution. We revise them with a robust MUSCL fallback scheme. The
linear advection equation is solved in both one and two dimensions and we
compare variations of these limited schemes based on their ability to maintain
a maximum principle, solution quality over long time integration and
computational cost.
  ...
