Erratum to: On the Stability of the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear Advection Equation
Erratum to: On the Stability of the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear Advection Equation
复制标题
勘误：关于线性平流方程四边形单元通量重建方案的稳定性
DOI：
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发表时间：
2016
影响因子：
2.5
通讯作者：
A. Jameson
中科院分区：
文献类型：
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作者：
A. Sheshadri;A. Jameson
高阶方法的通量重建（FR）方法已被证明是传统间断Galerkin（DG）方案的一个很有前途的替代方案，因为它们有助于采用适合于GPU等并行架构的显式时间步长方法。FR方法提供了一个参数化的家庭的计划，通过它可以恢复各种经典的计划，如双DG和谱差分方法。此外，可以改变参数以获得具有最大稳定时间步长或最小色散或耗散误差等的方案，为我们提供了一个统一高阶不连续有限元方法的单一的强大框架。已经有各种研究的准确性和稳定性，这些计划，特别是一个子集的FR计划被称为ESFR或VCJH计划已被证明是稳定的，在一维和单形元素在二维和三维的线性平流以及平流-扩散方程。然而，FR格式在张量积四边形单元上的稳定性一直是一个悬而未决的问题。虽然它是一维FR方法的最自然的扩展，但它提出了一个重大的挑战，特别是对于一般的四边形单元。本文研究了直角四边形网格上线性平流的VCJH型FR格式的稳定性，并证明了在一定条件下该格式可能变得不稳定。然而，我们发现恢复DG方法的VCJH格式在所有笛卡尔网格上都是稳定的。虽然我们限制自己的笛卡尔网格，以规避代数复杂性的雅可比矩阵的变化内一般张量积四边形元素，我们的分析提供了显着的洞察力，在一般四边形的FR方法的不稳定性的可能来源。
The flux reconstruction (FR) approach to high-order methods has proved to be a promising alternative to traditional discontinuous Galerkin (DG) schemes since they facilitate the adoption of explicit time-stepping methods suitable for parallel architectures like GPUs. The FR approach provides a parameterized family of schemes through which various classical schemes like nodal-DG and spectral difference methods can be recovered. Further, the parameters can be varied to obtain schemes with a maximum stable time-step, or minimum dispersion or dissipation errors etc., providing us a single powerful framework unifying high-order discontinuous Finite Element Methods. There have been various studies on the accuracy and the stability of these schemes and in particular, a subset of the FR schemes known as ESFR or VCJH schemes have been shown to be stable in 1D and on simplex elements in 2D and 3D for the linear advection as well as the advection---diffusion equations. However, the stability of the FR schemes on tensor product quadrilateral elements has remained an open question. Although it is the most natural extension of the 1D FR approach, it has posed a significant challenge, especially for general quadrilateral elements. In this paper, we investigate the stability of the VCJH-type FR schemes for linear advection on Cartesian quadrilateral meshes and show that the schemes could become unstable under certain conditions. However, we find that the VCJH scheme recovering the DG method is stable on all Cartesian meshes. Although we restrict ourselves to Cartesian meshes in order to circumvent the algebraic complexity posed by the variation of the Jacobian matrix inside general tensor-product quadrilateral elements, our analysis offers significant insight into the possible origins of instability in the FR approach on general quadrilaterals.