Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier–Stokes equations

David C. Del Rey Fernández, Mark H. Carpenter, Lisandro Dalcin, Stefano Zampini and Matteo Parsani ()
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David C. Del Rey Fernández: NASA Langley Research Center
Mark H. Carpenter: NASA Langley Research Center
Lisandro Dalcin: King Abdullah University of Science and Technology (KAUST)
Stefano Zampini: King Abdullah University of Science and Technology (KAUST)
Matteo Parsani: King Abdullah University of Science and Technology (KAUST)

Partial Differential Equations and Applications, 2020, vol. 1, issue 2, 1-54

Abstract: Abstract In this paper, we extend the entropy conservative/stable algorithms presented by Del Rey Fernández et al. (2019) for the compressible Euler and Navier–Stokes equations on nonconforming p-refined/coarsened curvilinear grids to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, we utilize a computationally simple and efficient approach based upon using decoupled interpolation operators. The resulting scheme is entropy conservative/stable and element-wise conservative. Numerical simulations of the isentropic vortex and viscous shock propagation confirm the entropy conservation/stability and accuracy properties of the method (achieving $$\sim p+1$$ ∼ p + 1 convergence), which are comparable to those of the original conforming scheme (Carpenter et al. in SIAM J Sci Comput 36(5):B835–B867, 2014; Parsani et al. in SIAM J Sci Comput 38(5):A3129–A3162, 2016). Simulations of the Taylor–Green vortex at $$\hbox {Re}=1600$$ Re = 1600 and turbulent flow past a sphere at $$\hbox {Re}_{\infty }=2000$$ Re ∞ = 2000 show the robustness and stability properties of the overall spatial discretization for unstructured grids. Finally, to demonstrate the entropy conservation property of a fully-discrete explicit entropy stable algorithm with h/p refinement/coarsening, we present the time evolution of the entropy function obtained by simulating the propagation of the isentropic vortex using a relaxation Runge–Kutta scheme.

Keywords: Nonconforming interfaces; h/p adaptation; Nonlinear entropy stability; Summation-by-parts; Simultaneous-approximation-terms; High-order accurate discretizations; Curved elements; Unstructured grid; 65M12; 65Z05; 76Nxx (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00009-z

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