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In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernández and coauthors [18,16,17] for the compressible Euler and Navier–Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, a computationally simple and efficient approach based upon using decoupled interpolation operators is utilized. 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 ∼ p + 1 convergence) which are comparable to those of the original conforming scheme [4,35]. Simulations of the Taylor–Green vortex at Re = 1, 600 and turbulent flow past a sphere at Re = 2, 000 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. David C. Del Rey Fernández National Institute of Aerospace and Computational AeroSciences Branch, NASA Langley Research Center, Hampton, VA, United States E-mail: dcdelrey@gmail.com Mark H. Carpenter NASA Langley Research Center, Hampton, VA, United States E-mail: mark.h.carpenter@nasa.gov Lisandro Dalcin, Stefano Zampini, Matteo Parsani King Abdullah University of Science and Technology (KAUST), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), Extreme Computing Research Center (ECRC), Thuwal, Saudi Arabia E-mail: dalcinl@gmail.com, stefano.zampini@kaust.edu.sa, matteo.parsani@kaust.edu.sa ar X iv :1 91 0. 02 11 0v 1 [ m at h. N A ] 4 O ct 2 01 9 2 David C. Del Rey Fernández et al.

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