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Entropy-Stable Multidimensional Summation-by-Parts Discretizations on hp -Adaptive Curvilinear Grids for Hyperbolic Conservation Laws
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-03-03 , DOI: 10.1007/s10915-020-01169-1
Siavosh Shadpey , David W. Zingg

Abstract

We develop high-order entropy-conservative semi-discrete schemes for hyperbolic conservation laws applicable to non-conforming curvilinear grids arising from h-, p-, or hp-adaptivity. More precisely, building on previous work with conforming grids by Crean et al. (J Comput Phys 356:410–438, 2018) and Chan et al. (SIAM J Sci Comput 41:A2938–A2966, 2019), we present two schemes: the first couples neighbouring elements in a skew-symmetric method, the second in a pointwise fashion. The key ingredients are degree p diagonal-norm summation-by-parts operators equipped with interface quadrature rules of degree 2p or higher, a skew-symmetric geometric mapping procedure using suitable polynomial functions, and a numerical flux that conserves mathematical entropy. Furthermore, entropy-stable schemes are obtained when augmenting the original schemes with a stabilization term that dissipates mathematical entropy at element interfaces. We provide both theoretical and numerical analysis for the compressible Euler equations demonstrating the element-wise conservation, entropy conservation/dissipation, and accuracy properties of the schemes. While both methods produce comparable results, our studies suggest that the scheme coupling elements in a pointwise manner is more computationally efficient.



中文翻译:

hp自适应双曲线守恒律的曲线网格上的熵稳定多维逐部分求和

摘要

针对双曲守恒律,我们开发了适用于由h-p-hp-适应性产生的非协调曲线网格的高阶熵守恒半离散方案。更准确地说,是在Crean等人以前的工作中采用一致的网格。(J Comput Phys 356:410–438,2018)和Chan等。(SIAM J Sci Comput 41:A2938–A2966,2019),我们提出了两种方案:第一种以偏斜对称方法耦合相邻元素,第二种以点方式耦合。关键要素是装备有2 p度接口正交规则的p角范数求和运算符或更高版本,使用合适的多项式函数的偏斜对称几何映射过程,以及可保留数学熵的数值通量。此外,当使用稳定项扩展原始方案时,可以获得熵稳定方案,该稳定项可以消除元素界面处的数学熵。我们为可压缩的Euler方程提供了理论和数值分析,论证了该方案的逐元素守恒,熵守恒/耗散以及精度性质。虽然这两种方法都能产生可比的结果,但我们的研究表明,以点方式耦合方案的方案在计算上更有效。

更新日期:2020-03-20
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