Search

In this paper, we extend the entropy conservative/stable algorithms presented by Del Rey Fernandez 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 $$\sim p+1$$ 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 $$\hbox {Re}=1600$$ and turbulent flow past a sphere at $$\hbox {Re}_{\infty }=2000$$ 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.