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arXiv 1902.01828math.NAcs.NA
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反对称熵稳定的模态间断Galerkin公式

Skew-symmetric entropy stable modal discontinuous Galerkin formulations

Jesse Chan

摘要:

求解非线性守恒律方程的高阶熵稳定间断Galerkin(DG)方法满足一个固有的离散熵不等式。这种计划的建设依赖于使用精心选择的节点或体积和表面求积规则,以产生满足部分求和(SBP)属性的运营商。在这项工作中,我们将展示如何构建“模态”斜对称DG配方是熵稳定的体积和表面求积下,传统的SBP属性不成立。这些反对称公式避免了使用“强”矩阵SBP属性,而是依赖于离散算子精确区分常数并满足微积分基本定理的离散形式的假设。最后,我们通过数值实验验证了所提出的公式的准确性和稳定性,并讨论了混合四边形-三角形网格上的熵稳定DG格式的反对称公式的应用。

英文摘要:

High order entropy stable discontinuous Galerkin (DG) methods for nonlinear conservation laws satisfy an inherent discrete entropy inequality. The construction of such schemes has relied on the use of carefully chosen nodal points or volume and surface quadrature rules to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct "modal" skew-symmetric DG formulations which are entropy stable for volume and surface quadratures under which a traditional SBP property does not hold. These skew-symmetric formulations avoid the use of a "strong" matrix-based SBP property, and instead rely on the assumption that discrete operators exactly differentiate constants and satisfy a discrete form of the fundamental theorem of calculus. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of skew-symmetric formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes.

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