Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements

JOURNAL OF COMPUTATIONAL PHYSICS

卷:356

Article

Crean, Jared¹  Hicken, Jason E.¹  Fernandez, David C. Del Rey³  Zingg, David W.²  Carpenter, Mark H.⁴ 

[1] Rensselaer Polytech Inst, Dept Mech Aerosp & Nucl Engn, Troy, NY 12181 USA

[2] Univ Toronto, Inst Aerosp Studies, Toronto, ON, Canada

[3] NIA, Hampton, VA USA

[4] NASA, Langley Res Ctr, Computat AeroSci Branch, Hampton, VA 23665 USA

关键词: Nonlinear entropy stability;    Summation-by-parts;    Simultaneous approximation terms;    High-order discretizations;    General elements;    Curved elements;    Unstructured grid;   

DOI  :  10.1016/j.jcp.2017.12.015

来源: Elsevier

【 摘 要 】

We present and analyze an entropy-stable semi-discretization of the Euler equations based on high-order summation-by-parts (SBP) operators. In particular, we consider general multidimensional SBP elements, building on and generalizing previous work with tensor-product discretizations. In the absence of dissipation, we prove that the semi-discrete scheme conserves entropy; significantly, this proof of nonlinear L-2 stability does not rely on integral exactness. Furthermore, interior penalties can be incorporated into the discretization to ensure that the total (mathematical) entropy decreases monotonically, producing an entropy-stable scheme. SBP discretizations with curved elements remain accurate, conservative, and entropy stable provided the mapping Jacobian satisfies the discrete metric invariants; polynomial mappings at most one degree higher than the SBP operators automatically satisfy the metric invariants in two dimensions. In three-dimensions, we describe an elementwise optimization that leads to suitable Jacobians in the case of polynomial mappings. The properties of the semi-discrete scheme are verified and investigated using numerical experiments. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free