Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs

In this article, we present a unified framework of discontinuous Galerkin (DG) discretizations for Hamiltonian ODEs and PDEs. We show that with appropriate numerical fluxes the numerical algorithms deduced from DG discretizations can be combined with the symplectic methods in time to derive the multi-symplectic PRK schemes. The resulting numerical discretizations are applied to the linear and nonlinear Schrodinger equations. Some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments.

[1]  Xinyuan Wu,et al.  Functionally Fitted Energy-Preserving Methods for Solving Oscillatory Nonlinear Hamiltonian Systems , 2016, SIAM J. Numer. Anal..

[2]  M. Borri,et al.  A general framework for interpreting time finite element formulations , 1993 .

[3]  P. Betsch,et al.  Inherently Energy Conserving Time Finite Elements for Classical Mechanics , 2000 .

[4]  Zhang Mei-Qing,et al.  Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations , 1990 .

[5]  R. D. Vogelaere,et al.  Methods of Integration which Preserve the Contact Transformation Property of the Hamilton Equations , 1956 .

[6]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[7]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[8]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[9]  Eitan Tadmor,et al.  Strong Stability-Preserving High-Order Time Discretization , 2001 .

[10]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[11]  Chi-Wang Shu,et al.  Local discontinuous Galerkin methods for nonlinear Schrödinger equations , 2005 .

[12]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[13]  Geng Sun,et al.  The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs , 2005, Math. Comput..

[14]  C. Bottasso.  A new look at finite elements in time: a variational interpretation of Runge-Kutta methods , 1997 .

[15]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives , 2002, J. Sci. Comput..

[16]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[17]  Robert I. McLachlan,et al.  High Order Multisymplectic Runge-Kutta Methods , 2014, SIAM J. Sci. Comput..

[18]  Wensheng Tang,et al.  Construction of Runge-Kutta type methods for solving ordinary differential equations , 2014, Appl. Math. Comput..

[19]  Wensheng Tang,et al.  A New Approach to Construct Runge-Kutta Type Methods and Geometric Numerical Integrators , 2012 .

[20]  Yajuan Sun,et al.  Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs , 2007, Numerische Mathematik.

[21]  C. Schober,et al.  Geometric integrators for the nonlinear Schrödinger equation , 2001 .

[22]  B. Hulme.  Discrete Galerkin and related one-step methods for ordinary differential equations , 1972 .

[23]  Wensheng Tang,et al.  Time finite element methods: A unified framework for numerical discretizations of ODEs , 2012, Appl. Math. Comput..

[24]  Robert I. McLachlan,et al.  On Multisymplecticity of Partitioned Runge-Kutta Methods , 2008, SIAM J. Sci. Comput..

[25]  Bin Wang,et al.  Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes , 2004, Appl. Math. Comput..

[26]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[27]  Jing-Bo Chen.  Multisymplectic geometry, local conservation laws and Fourier pseudospectral discretization for the "good" Boussinesq equation , 2005, Appl. Math. Comput..

[28]  Jialin Hong.  A Survey of Multi-symplectic Runge-Kutta Type Methods for Hamiltonian Partial Differential Equations , 2006 .

[29]  Wensheng Tang,et al.  Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage , 2015, Appl. Math. Comput..

[30]  汤琼,et al.  Continuous finite element methods for Hamiltonian systems , 2007 .

[31]  E. Hairer.  Energy-preserving variant of collocation methods 1 , 2010 .

[32]  Geng.  CONSTRUCTION OF HIGH ORDER SYMPLECTIC PRK METHODS , 1995 .