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

Author

Listed:
  • 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))

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.

Suggested Citation

  • David C. Del Rey Fernández & Mark H. Carpenter & Lisandro Dalcin & Stefano Zampini & Matteo Parsani, 2020. "Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier–Stokes equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-54, April.
  • Handle: RePEc:spr:pardea:v:1:y:2020:i:2:d:10.1007_s42985-020-00009-z
    DOI: 10.1007/s42985-020-00009-z
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    References listed on IDEAS

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    1. Gassner, Gregor J. & Winters, Andrew R. & Kopriva, David A., 2016. "A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 291-308.
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