Relative Efficiency of Finite-Difference and Discontinuous Spectral-Element Summation-by-Parts Methods on Distorted Meshes

Conventional wisdom suggests that high-order finite-difference methods are more efficient than high-order discontinuous spectral-element methods on smooth meshes, but less efficient as the mesh becomes increasingly distorted because of a significant loss of accuracy on such meshes. This paper investigates the influence of mesh distortion on the relative efficiency of different implementations of generalized summation-by-parts (GSBP) methods, with emphasis on comparing finite-difference and discontinuous spectral-element approaches. These include discretizations built using classical finite-difference SBP operators, with and without optimized boundary closures, as well as both Legendre-Gauss and Legendre-Gauss-Lobatto operators. The traditionally finite-difference operators are also applied as discontinuous spectral-element operators by selecting a fixed number of nodes per element and performing mesh refinement by increasing the number of elements rather than the number of mesh nodes. Using the linear convection equation and nonlinear Euler equations as models, solutions are obtained on meshes with different types and severity of distortion. Contrary to expectation, the results show that finite-difference implementations are no more sensitive to mesh distortion than discontinuous spectral-element implementations, maintaining their relative efficiency in most cases. The results also show that the operators of Mattsson et al. (J Comput Phys 264:91–111, 2014) with optimized boundary operators are often the most efficient for a given implementation strategy (finite-difference or discontinuous spectral-element). While their accuracy as finite-difference operators might be expected, their superior accuracy to LG and LGL nodal distributions when implemented as discontinuous spectral-element operators is not well known.