High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere.

*(English)*Zbl 1089.65096Summary: High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2\(N\) Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented by F. X. Giraldo and T. Warburton [ibid. 207, No. 1, 129–150 (2005; Zbl 1177.86002)], is small, local and efficient to invert.

Two types of finite volume-type flux integrals are studied: a set based on Gauss-Lobatto quadrature points (order 2\(N-1)\) and a set based on Gauss quadrature points (order 2\(N\)). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with \(2N\) integration being the most accurate of the four DG methods studied. The strong advection form with \(2N\) integration performed extremely well even for flows with shock waves. The strong conservation form with 2\(N - 1\) integration yielded results almost as good as those with 2\(N\) while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

Two types of finite volume-type flux integrals are studied: a set based on Gauss-Lobatto quadrature points (order 2\(N-1)\) and a set based on Gauss quadrature points (order 2\(N\)). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with \(2N\) integration being the most accurate of the four DG methods studied. The strong advection form with \(2N\) integration performed extremely well even for flows with shock waves. The strong conservation form with 2\(N - 1\) integration yielded results almost as good as those with 2\(N\) while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76M12 | Finite volume methods applied to problems in fluid mechanics |

76M10 | Finite element methods applied to problems in fluid mechanics |

##### Keywords:

finite element; Finite volume; numerical examples; comparison of methods; conservation law; discontinuous Galerkin methods; hyperbolic equations; rotating sphere; spectral element method; Gauss quadrature points; Gauss quadrature
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\textit{F. X. Giraldo}, J. Comput. Phys. 214, No. 2, 447--465 (2006; Zbl 1089.65096)

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