A novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high‐order local polynomial‐based framework, for example, the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes. The high‐order polynomial approximation generally fails to correctly capture a strong discontinuity inside a cell due to the Gibbs phenomenon, which is replaced by more stable approximation on the basis of a troubled‐cell indicator such as that used with the total variation bounded (TVB) limiter. This paper examines the applicability of a new algorithm, so‐called boundary variation diminishing (BVD) reconstruction, to the weighted essentially nonoscillatory methodology in the FR framework including the nodal type DG method. The BVD reconstruction adaptively chooses a proper approximation for the solution function so as to minimize the jump between values at the left and right side of cell boundaries. The results of the BVD algorithm are comparable to those with the conventional TVB limiter in terms of oscillation suppression and numerical dissipation in one‐dimensional linear advection and nonlinear system equations, while the TVB limiter performs better in the case with strong discontinuities (the blast wave problem in the Euler equations). Overall, since the present BVD algorithm does not need any ad hoc constant such as the TVB parameter, it could be more reliable than the conventional TVB limiter that is often used in the DG and FR communities for shocks and other discontinuities. The proposed method would lead to a parameter‐free robust algorithm for the local polynomial‐based high‐order schemes.

中文翻译：

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高阶局部多项式方案的边界变化递减算法

在基于高阶局部多项式的框架中，针对双曲守恒律采用了一种选择适当重构的新颖方法，例如，非连续Galerkin（DG）和通量重构（FR）方案。由于Gibbs现象，高阶多项式逼近通常无法正确捕获像元内部的强不连续性，在有问题的像元指标（例如与总变化有界（TVB）一起使用的指标）的基础上，高阶多项式逼近被更稳定的逼近所替代。 ）限制器。本文研究了一种新的算法，即所谓的边界变化递减（BVD）重构，是否适用于FR框架（包括节点类型DG方法）中的加权基本非振荡方法。BVD重建为求解函数自适应地选择一个适当的近似值，以最小化单元格边界左侧和右侧的值之间的跳变。在一维线性对流和非线性系统方程中，BVD算法的结果在振动抑制和数值耗散方面可与常规TVB限幅器相媲美，而在强烈不连续性（爆炸波）的情况下，TVB限幅器的性能更好欧拉方程中的问题）。总体而言，由于当前的BVD算法不需要诸如TVB参数之类的任何临时常数，因此它比在DG和FR社区中经常用于冲击和其他不连续性的常规TVB限制器更可靠。