# American Institute of Mathematical Sciences

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

## A local discontinuous Galerkin method based on variational structure

 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

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.
Citation: Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & 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. 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. 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. 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, (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. doi: 10.1137/S0036142997316712. [6] A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation,, Physica D, 134 (1999), 200. 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. 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, (2011). [9] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.),, Springer-Verlag, (2006). [10] O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449. 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. 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. 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). 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. doi: 10.1137/S0036142997316712. [6] A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation,, Physica D, 134 (1999), 200. 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. 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, (2011). [9] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.),, Springer-Verlag, (2006). [10] O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449. 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. 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. 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). 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, (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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jcp.2013.01.031.
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