High-Order Summation-By-Parts Schemes for First-Order Hyperbolic Systems in Curvilinear Coordinates with Singularities
This work addresses the challenge of stable numerical methods for hyperbolic systems in curvilinear coordinates with singularities, which is important for computational physics and engineering applications involving spherical geometries.
The paper presents a method for constructing high-order accurate, energy-stable finite difference operators satisfying the Summation-by-Parts property on spherical domains with coordinate singularities, up to order six. Example evolutions of the scalar wave equation demonstrate the advantages of these operators.
Formulating stable numerical methods for hyperbolic systems in curvilinear coordinate 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 [C. Gundlach, J. M. Martín-García, and D. Garfinkle, CQG 30, 145003 (2013)]. We define discrete gradient and divergence operators that mirror the continuous integration-by-parts principle, even though there is a $1/r^p$ 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.