Numerical investigation of tensor-product summation-by-parts discretization strategies and operators

This paper presents a numerical investigation of the tradeoffs between various discretization approaches and operators, based on diagonal-norm summation-by-parts (SBP) operators, using the two-dimensional linear convection equation and simultaneous approximation terms (SATs) for the weak imposition of boundary conditions and interface coupling. In particular, it focuses on operators which include boundary nodes. Of the operators considered, the hybrid-Gauss-trapezoidal-Lobatto SBP operators are the most efficient. Little difference in efficiency is observed between the divergence and skew-symmetric forms, making the latter preferred given its provable stability on curved meshes. The traditional finite-difference refinement strategy is the most efficient, and the discontinuous element approach the least. The continuous element refinement strategy has comparable efficiency to the traditional approach when not exhibiting lower convergence rates. This motivates a hybrid approach whereby discontinuous elements are constructed from continuous subelements. This hybrid approach is found to inherit the higher convergence rates of the traditional and discontinuous approaches, and higher efficiency relative to the discontinuous approach.