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

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Abstract

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.

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Data Availability

The authors declare that the data supporting the findings of this study are available within the paper and its supplementary files in graphical form. Raw data can be provided from the corresponding author on reasonable request.

Notes

  1. We expect that three-dimensional results will exhibit similar trends.

  2. For first derivative operators, the order and polynomial degree of the operator are the same.

  3. The values presented here are rounded to 3 decimal places for the sake of space. Please refer to [14] for higher precision values of the operator’s coefficients.

  4. Figures show the square root of the number of nodes; accounting for the simulations being in two dimensions. This is to enable reference convergence slopes to be added to the figures.

  5. The term cell is used here to indicate the quadrilateral formed by 4 nodes in a mesh with indices (ij), \((i+1,j)\), \((i+1,j+1)\), and \((i,j+1)\) - not a spectral-element.

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Boom, P.D., Del Rey Fernández, D.C. & Zingg, D.W. Relative Efficiency of Finite-Difference and Discontinuous Spectral-Element Summation-by-Parts Methods on Distorted Meshes. J Sci Comput 102, 28 (2025). https://doi.org/10.1007/s10915-024-02745-5

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