American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations,, Chapman & Hall/CRC, (2011).   Google Scholar

[9]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.),, Springer-Verlag, (2006).   Google Scholar

[10]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation,, Technical report, (1973).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations,, Chapman & Hall/CRC, (2011).   Google Scholar

[9]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.),, Springer-Verlag, (2006).   Google Scholar

[10]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Sci., 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation,, Technical report, (1973).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206
[2]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146
[3]

Lin Lu, Qi Wang, Yongzhong Song, Yushun Wang. Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020311
[4]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018
[5]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
[6]

Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic & Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373
[7]

Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021
[8]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[9]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[10]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[11]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, , () : -. doi: 10.3934/era.2020078
[12]

Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602
[13]

Hassan Najafi Alishah, Pedro Duarte, Telmo Peixe. Conservative and dissipative polymatrix replicators. Journal of Dynamics & Games, 2015, 2 (2) : 157-185. doi: 10.3934/jdg.2015.2.157
[14]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[15]

Alexander J. Zaslavski. Structure of approximate solutions of Bolza variational problems on large intervals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1283-1316. doi: 10.3934/dcdss.2018072
[16]

Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405
[17]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[18]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[19]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[20]

Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences