Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations and equations of transport type. Limiting is required to either converge to the correct type of solution or to adhere to physically motivated local maximum principles. Less restrictive limiting procedures are required so as to not severely decrease the accuracy. In this paper, we develop an existing slope limiter framework, to achieve different local boundedness principles for higher-order schemes on unstructured meshes. Quadrature points contributing to numerical fluxes can be limited based on face defined maximum principles, and the resulting cell mean at the next timestep can satisfy a cell mean maximum principle but with less limiting. We demonstrate the practical application of the introduced framework to a second-order finite volume scheme as well as a fourth-order finite volume scheme, in the context of the advection equation.

翻译：高阶数值方法用于求解双曲型偏微分方程及输运型方程的精确数值解。为确保收敛至正确解类型或满足物理驱动的局部最大值原理，需采用限制技术。为不过度降低精度，需采用限制性较小的限制过程。本文在现有斜率限制器框架基础上进行扩展，为非结构网格上的高阶格式实现不同的局部有界性原理。参与数值通量计算的积分点可根据面定义的最大值原理进行限制，从而使得下一时间步的单元均值在满足单元均值最大值原理的同时减少限制程度。以对流方程为例，我们展示了所提框架在二阶有限体积格式及四阶有限体积格式中的实际应用。