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High-order summation-by-parts schemes for first-order hyperbolic systems in curvilinear coordinates with singularities

Stamatis Vretinaris*

Erik Schnetter

  • *Contact author: stamatis.vretinaris@aei.mpg.de, stamatis.vretinaris@ru.nl
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Phys. Rev. D 114, 084089 – Published 28 October, 2026

DOI: https://doi.org/10.1103/hg1v-qxsj

Abstract

Formulating stable numerical methods for hyperbolic systems in curvilinear coordinates with singularities, e.g., spherical coordinates, is complicated by the presence of these singularities. We present a method for constructing high-order accurate, energy-stable finite difference operators satisfying the summation by parts (SBP) property on spherical domains, extending ideas presented by Gundlach, Martín-García, and Garfinkle [1]. We define discrete gradient and divergence operators that mirror the continuous integration-by-parts principle, even though there is a 1/rp coordinate singularity present at the origin. We explicitly construct such operators up to order six. Our operators place a grid point directly on the origin. We also review how to construct stable SBP operators that straddle the origin. We analyze the accuracy and spectral radii of these operators, and we show example evolutions of the scalar wave equation to demonstrate the advantages of such operators.

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References (39)

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Phys. Rev. D 114, 084089– Published 28 October, 2026
  • Received 8 June 2026
  • Accepted 25 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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