Boundary variation diminishing algorithm for high‐order local polynomial‐based schemes
Boundary variation diminishing algorithm for high‐order local polynomial‐based schemes
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DOI：
10.1002/fld.4899
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发表时间：
2020-07
影响因子：
1.8
通讯作者：
Yoshiaki Abe;Ziyao Sun;F. Xiao
中科院分区：
文献类型：
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作者：
Yoshiaki Abe;Ziyao Sun;F. Xiao
在基于高阶局部多项式的框架下，对双曲型守恒律进行了一种新的重构方法，例如间断Galerkin（DG）和通量重构（FR）格式.由于吉布斯现象，高阶多项式近似通常无法正确捕获单元内的强不连续性，这被基于故障单元指标（例如与总变差有界（TVB）限制器一起使用的指标）的更稳定的近似所取代。本文研究了一种新的算法，即所谓的边界变差递减（BVD）重建，在FR框架中的加权基本无振荡方法，包括节点型DG方法的适用性。BVD重建自适应地选择解函数的适当近似，以最小化单元边界的左侧和右侧的值之间的跳跃。在一维线性平流和非线性系统方程的振荡抑制和数值耗散方面，BVD算法的结果与传统的TVB限制器相当，而TVB限制器在强间断情况下（欧拉方程中的爆炸波问题）表现得更好。总的来说，由于本BVD算法不需要任何特设常数，如TVB参数，它可能是更可靠的比传统的TVB限制器，经常使用在DG和FR社区的冲击和其他不连续性。所提出的方法将导致无参数的鲁棒算法的局部多项式为基础的高阶格式。
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.