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Discontinuous Galerkin and Related Methods for ODE

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Abstract

A defining feature of the discontinuous Galerkin (DG) method for ODE is that the piecewise polynomial solution can have a jump discontinuity at the beginning of each step. Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is shown to be identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge–Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG via correction function also clarifies the meaning and facilitates the proofs of various \(B(p)\), \(C(\eta )\), and \(D(\zeta )\) conditions for accuracy. The two consequences of these conditions are: all \(s\)-stage IRK-DG methods are accurate to order \(2s-1\), and the IRK-DG methods of Radau type are unique. L-stability of the IRK-DG method is discussed. In all, the correction function plays a key role and helps establish the relations among the DG, IRK-DG, collocation, and CG schemes.

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Acknowledgements

The author was supported by the Transformational Tools and Technologies Project of NASA. He also wishes to thank Dr. Dimitri Mavriplis for several interesting discussions on DG methods applied to time stepping and Dr. Seth Spiegel as well as an unknown reviewer for their thorough reviews and numerous valuable comments and suggestions.

Funding

This work was supported by the Transformational Tools and Technologies Project of NASA.

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  1. NASA Glenn Research Center, MS 5 – 11, Cleveland, OH, 44135, USA

    H. T. Huynh

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Huynh, H.T. Discontinuous Galerkin and Related Methods for ODE. J Sci Comput 96, 51 (2023). https://doi.org/10.1007/s10915-023-02233-2

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