Entropy-Stable Multidimensional Summation-by-Parts Discretizations on hp-Adaptive Curvilinear Grids for Hyperbolic Conservation Laws

We develop high-order entropy-conservative semi-discrete schemes for hyperbolic conservation laws applicable to non-conforming curvilinear grids arising from h-, p-, or hp-adaptivity. More precisely, building on previous work with conforming grids by Crean et al. (J Comput Phys 356:410–438, 2018) and Chan et al. (SIAM J Sci Comput 41:A2938–A2966, 2019), we present two schemes: the first couples neighbouring elements in a skew-symmetric method, the second in a pointwise fashion. The key ingredients are degree p diagonal-norm summation-by-parts operators equipped with interface quadrature rules of degree 2p or higher, a skew-symmetric geometric mapping procedure using suitable polynomial functions, and a numerical flux that conserves mathematical entropy. Furthermore, entropy-stable schemes are obtained when augmenting the original schemes with a stabilization term that dissipates mathematical entropy at element interfaces. We provide both theoretical and numerical analysis for the compressible Euler equations demonstrating the element-wise conservation, entropy conservation/dissipation, and accuracy properties of the schemes. While both methods produce comparable results, our studies suggest that the scheme coupling elements in a pointwise manner is more computationally efficient.