A local discontinuous Galerkin method based on variational structure

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October 2015, 8(5): 817-832. doi: 10.3934/dcdss.2015.8.817

A local discontinuous Galerkin method based on variational structure

Yoshifumi Aimoto ^1, , Takayasu Matsuo ^1, and Yuto Miyatake ^1,

1.	Department of Mathematical Informatics, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-0033, Japan, Japan, Japan

Received December 2013 Revised June 2014 Published July 2015

Abstract
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We present a special variant of the local discontinuous Galerkin (LDG) method for time-dependent partial differential equations with certain variational structures and associated conservation or dissipation properties. The method provides a way to construct fully-discrete LDG schemes that retain discrete counterparts of the conservation or dissipation properties. Numerical results confirm the accuracy and effectiveness of the method.

Keywords: conservative, structure-preserving, Discontinuous Galerkin method, dissipative., variational structure.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817

References:

[1]	F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572.
[2]	J. L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comput., 82 (2013), 1401-1432. doi: 10.1090/S0025-5718-2013-02661-0.
[3]	G. F. Carey and Y. Shen, Approximations of the KdV equation by least squares finite elements, Comput. Methods Appl. Mech. Engrg., 93 (1991), 1-11. doi: 10.1016/0045-7825(91)90112-J.
[4]	B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin methods, Theory, Computation and Applications, volume 11 of Springer Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-642-59721-3.
[5]	B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for timedependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.
[6]	A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D, 134 (1999), 200-226. doi: 10.1016/S0167-2789(99)00072-X.
[7]	D. Furihata, Finite difference schemes for $\frac{\partial u}{\partial t}=(\frac{\partial}{\partial x})^\alpha\frac{\delta G}{\delta u}$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377.
[8]	D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, 2011.
[9]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Springer-Verlag, Heidelberg, 2006.
[10]	O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.
[11]	T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations, J. Comput. Appl. Math., 218 (2008), 506-521. doi: 10.1016/j.cam.2007.08.001.
[12]	Y. Miyatake and T. Matsuo, A general framework for finding energy dissipative/conservative $H^1$-Galerkin schemes and their underlying $H^1$-weak forms for nonlinear evolution equations, BIT., 54 (2014), 1119-1154. doi: 10.1007/s10543-014-0483-3.
[13]	G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.
[14]	W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Technical report, Los Alamos Scientific Laboratory Report, 1973.
[15]	Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, J. Comput. Phys., 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.
[16]	Y. Xing, C. S. Chou and C. W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problem and Imaging, 7 (2013), 967-986. doi: 10.3934/ipi.2013.7.967.
[17]	Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72-97. doi: 10.1016/j.jcp.2004.11.001.
[18]	Y. Xu and C. W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), 1998-2021. doi: 10.1137/070679764.
[19]	T. Yaguchi, T. Matsuo and M. Sugihara, An extension of the discrete variational method to nonuniform grids, J. Comput. Phys., 229 (2010), 4382-4423. doi: 10.1016/j.jcp.2010.02.018.
[20]	J. Yan and C. W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), 769-791. doi: 10.1137/S0036142901390378.
[21]	N. Yi, Y. Huang and H. Liu, A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: Energy conservation and boundary effect, J. Comput. Phys., 242 (2013), 351-366. doi: 10.1016/j.jcp.2013.01.031.

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References:

[1]	F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572.
[2]	J. L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comput., 82 (2013), 1401-1432. doi: 10.1090/S0025-5718-2013-02661-0.
[3]	G. F. Carey and Y. Shen, Approximations of the KdV equation by least squares finite elements, Comput. Methods Appl. Mech. Engrg., 93 (1991), 1-11. doi: 10.1016/0045-7825(91)90112-J.
[4]	B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin methods, Theory, Computation and Applications, volume 11 of Springer Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-642-59721-3.
[5]	B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for timedependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.
[6]	A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D, 134 (1999), 200-226. doi: 10.1016/S0167-2789(99)00072-X.
[7]	D. Furihata, Finite difference schemes for $\frac{\partial u}{\partial t}=(\frac{\partial}{\partial x})^\alpha\frac{\delta G}{\delta u}$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377.
[8]	D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, 2011.
[9]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Springer-Verlag, Heidelberg, 2006.
[10]	O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.
[11]	T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations, J. Comput. Appl. Math., 218 (2008), 506-521. doi: 10.1016/j.cam.2007.08.001.
[12]	Y. Miyatake and T. Matsuo, A general framework for finding energy dissipative/conservative $H^1$-Galerkin schemes and their underlying $H^1$-weak forms for nonlinear evolution equations, BIT., 54 (2014), 1119-1154. doi: 10.1007/s10543-014-0483-3.
[13]	G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7pp. doi: 10.1088/1751-8113/41/4/045206.
[14]	W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Technical report, Los Alamos Scientific Laboratory Report, 1973.
[15]	Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, J. Comput. Phys., 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.
[16]	Y. Xing, C. S. Chou and C. W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problem and Imaging, 7 (2013), 967-986. doi: 10.3934/ipi.2013.7.967.
[17]	Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72-97. doi: 10.1016/j.jcp.2004.11.001.
[18]	Y. Xu and C. W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), 1998-2021. doi: 10.1137/070679764.
[19]	T. Yaguchi, T. Matsuo and M. Sugihara, An extension of the discrete variational method to nonuniform grids, J. Comput. Phys., 229 (2010), 4382-4423. doi: 10.1016/j.jcp.2010.02.018.
[20]	J. Yan and C. W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), 769-791. doi: 10.1137/S0036142901390378.
[21]	N. Yi, Y. Huang and H. Liu, A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: Energy conservation and boundary effect, J. Comput. Phys., 242 (2013), 351-366. doi: 10.1016/j.jcp.2013.01.031.

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