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

收稿日期:2021-04-30 修回日期:2022-06-27 发布日期:2023-12-16
通讯作者: 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
基金资助:
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.

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.

摘要/Abstract

摘要： 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.

关键词: Discontinuous Galerkin, Fourier continuation(FC), High order method

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

中图分类号:

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.

参考文献

1. Ainsworth, M.: Dispersive and dissipative behavior of high-order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106-130 (2004)
2. Albin, N., Bruno, O.P.: A spectral FC solver for the compressible Navier-Stokes equations in general domains I: explicit time-stepping. J. Comput. Phys. 230(16), 6248-6270 (2011)
3. Albin, N., Bruno, O.P., Cheung, T.Y., Cleveland, R.O.: Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams. J. Acoust. Soc. Am. 132(4), 2371-2387 (2012)
4. Albin, N., Pathmanathan, S.: Discrete periodic extension using an approximate step function. SIAM J. Sci. Comput. 36(2), A668-A692 (2014)
5. Appelö, D., Bokil, V.A., Cheng, Y., Li, F.: Energy stable SBP-FDTD methods for Maxwell-Duffing models in nonlinear photonics. IEEE J. Multiscale Multiphys. Comput. Tech. 4, 329-336 (2019)
6. Boyd, J.P.: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178(1), 118-160 (2002)
7. Bruno, O.P., Lyon, M.: High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements. J. Comput. Phys. 229(6), 2009-2033 (2010)
8. Bruno, O.P., Prieto, A.: Spatially dispersionless, unconditionally stable FC-AD solvers for variablecoefficient PDEs. J. Sci. Comput. 58, 331-366 (2014)
9. Cockburn, B., Shu, C-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411-435 (1989)
10. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173-261 (2001)
11. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2013)
12. Fornberg, B., Reeger, J.: An improved Gregory-like method for 1-D quadrature. Numer. Math. 141, 1-19 (2019)
13. Gustafsson, B., Kreiss, H., Oliger, J.: Time Dependent Problems and Difference Methods. Pure and Applied Mathematics. Wiley, New York (1995)
14. Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids—I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181(1), 186-221 (2002)
15. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, vol. 54. Springer, New York (2008)
16. Huybrechs, D.: On the Fourier extension of nonperiodic functions. SIAM J. Numer. Anal. 47(6), 4326-4355 (2010)
17. Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191(1), 249-264 (2003)
18. Kopriva, D.A.: Stability of overintegration methods for nodal discontinuous Galerkin spectral element methods. J. Sci. Comput. 76(1), 426-442 (2018)
19. Kronbichler, M., Kormann, K.: Fast matrix-free evaluation of discontinuous Galerkin finite element operators. ACM Trans. Math. Softw. 45(3), 1-40 (2019)
20. Lyon, M.: A fast algorithm for Fourier continuation. SIAM J. Sci. Comput. 33, 3241-3260 (2011)
21. Lyon, M., Bruno, O.P.: High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations. J. Comput. Phys. 229, 3358- 3381 (2010)
22. Persson, P.O.: A sparse and high-order accurate line-based discontinuous Galerkin method for unstructured meshes. J. Comput. Phys. 233, 414-429 (2013)

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Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

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