SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Local discontinuous Galerkin methods for Hamiltonian PDEs

Y. Sun, W. Cai and W. Tang

Abstract
In this talk, we present the time-space local discontinuous Galerkin methods [1] for Hamiltonian PDEs in the multisymplectic formulation. With the appropriate quadrature formula, the resulting discretizations are equivalent to the multisymplectic partitioned Runge-Kutta methods [2] whose implementation has not been fully understood besides for a class of multisymplectic Hamiltonian systems in special forms [3]. We use the new discretization strategy to simulate the soliton solutions of nonlinear Schrödinger equation, and observe the errors of numerical solutions and the global charge.

Bibliography
[1] Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72-97.
[2] J. Hong, H. Liu and G. Sun, The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs, Math. Comput., 75(253) (2005), 167-181.
[3] B. N. Ryland and R. I. McLachlan, On multisymplecticity of partitioned Runge-Kutta methods, SIAM J. Sci. Comput. 30(3) (2008), 1318-1340.

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