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A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation

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Abstract

In this paper we introduce a hybridized discontinuous Galerkin (HDG) method for solving nonlinear Korteweg–de Vries type equations. Similar to a standard HDG implementation, we first express the approximate variables and numerical fluxes inside each element in terms of the approximate traces of the scalar variable (u), and its first derivative (\(u_x\)). These traces are assumed to be single-valued on each face. Next, we impose the conservation of numerical fluxes via two extra sets of equations. Using these global flux conservation conditions and applying the Newton–Raphson method, we construct a system of equations that can be solely expressed in terms of the increments of approximate traces in each iteration. Afterwards, we solve these equations, and substitute the approximate traces back into local equations over each element to obtain local approximate solutions. As for the time stepping scheme, we use the backward difference formulae. The method is proved to be stable for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with kth order elements, the computed u, p, and q show optimal convergence at order \(k+1\) in both linear and nonlinear cases, which improves upon previously employed techniques.

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Acknowledgments

The authors would like to acknowledge the support of National Science Foundation Grant ACI 1339801. We also want to express our gratitude to the anonymous reviewers for their valuable comments to improve the quality of this paper.

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Authors and Affiliations

  1. Computational Hydraulics Group (CHG), Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX, 78712, USA

    Ali Samii, Craig Michoski & Clint Dawson

  2. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 78712, USA

    Ali Samii & Clint Dawson

  3. Department of Statistics, Colorado State University, Fort Collins, CO, 80523, USA

    Nishant Panda

Authors
  1. Ali Samii
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  2. Nishant Panda
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  3. Craig Michoski
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Correspondence to Ali Samii.

Additional information

The authors acknowledge the support of National Science Foundation Grant ACI 1339801.

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Samii, A., Panda, N., Michoski, C. et al. A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation. J Sci Comput 68, 191–212 (2016). https://doi.org/10.1007/s10915-015-0133-1

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  • DOI: https://doi.org/10.1007/s10915-015-0133-1

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