Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

doi:10.1007/s42967-022-00205-1

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1385-1405.doi: 10.1007/s42967-022-00205-1

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Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Kiera van der Sande¹, Daniel Appelö^2,3, Nathan Albin⁴

1 Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO, USA;
2 Department of Computational Mathematics, Science & Engineering, Michigan State University, East Lansing, USA;
3 Department of Mathematics, Michigan State University, East Lansing, USA;
4 Department of Mathematics, Kansas State University, Manhattan, KS, USA

Received:2021-04-30 Revised:2022-06-27 Published:2023-12-16
Contact: Daniel Appelö,E-mail:appeloda@msu.edu;Kiera van der Sande,E-mail:kiera.vandersande@colorado.edu;Nathan Albin,E-mail:albin@k-state.edu E-mail:appeloda@msu.edu;kiera.vandersande@colorado.edu;albin@k-state.edu
Supported by:
This work was supported by the National Science Foundation Grant DMS-1913076. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Abstract

Abstract: Fourier continuation (FC) is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions. These methods have been used in partial differential equation (PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving the stability and the convergence. Here we propose the use of FC in forming a new basis for the DG framework.

Key words: Discontinuous Galerkin, Fourier continuation(FC), High order method

CLC Number:

65M60
65M06

Kiera van der Sande, Daniel Appelö, Nathan Albin. Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1385-1405.

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References

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Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

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