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. ...

翻译：求解含光滑解的双曲型守恒律时，高阶有限体积法和有限元法在精度与计算效率上表现出色。然而，当解包含间断时，此类高阶方法会引入非物理振荡及严重的过冲/下冲现象。斜率限制器作为有效补救措施，通过保持单调性抑制振荡，部分限制器甚至能在数值解中维持严格的最大值原理。限制器可分为两类：先验限制器与后验限制器。前者仅基于当前时刻$t^n$的数据修正高阶解，后者则需计算$t^{n+1}$时刻的候选解，并迭代重新计算直至满足特定条件。这两种限制范式均适用于有限体积法和有限元法。本研究提出一种方法论，用于比较任意高阶有限体积求解器中先验与后验限制器的性能。我们选取文献\cite{zhang2011maximum, zhang2010maximum}中提出的最大值原理保持格式作为先验限制格式，后验限制则采用文献\cite{clain2011high}的方法，在候选解中识别所谓的"扰动单元"，并通过稳健的MUSCL回退格式进行修正。通过求解一维与二维线性对流方程，我们从最大值原理保持能力、长时间积分解质量及计算成本三个方面对这些限制格式的变体进行比较。