Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs

ABSTRACT

This paper examines the novel local discontinuous Galerkin (LDG) discretization for Hamiltonian PDEs based on its multisymplectic formulation. This new kind of LDG discretizations possess one major advantage over other standard LDG method, which, through specially chosen numerical fluxes, states the preservation of discrete conservation laws (i.e. energy), and also the multisymplectic structure while the symplectic time integration is adopted. Moreover, the corresponding local multisymplectic conservation law holds at the units of elements instead of each node. Taking the nonlinear Schrödinger equation and the KdV equation as the examples, we illustrate the derivations of discrete conservation laws and the corresponding numerical fluxes. Numerical experiments by using the modified LDG method are demonstrated for the sake of validating our theoretical results.

KEYWORDS:

• Multisymplectic formulation
• Hamiltonian PDEs
• local discontinuous Galerkin method
• numerical flux
• conservation law

2010 AMS SUBJECT CLASSIFICATIONS:

• 35L65
• 37K05
• 65M20
• 65M22
• 65M60

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

W. Cai http://orcid.org/0000-0002-6149-3780

Notes

1 Let $\mathcal{N}$ be an operator on the Banach space. The symmetry condition means ${\mathcal{N}}_u^{\mathrm{\prime }}={\tilde{\mathcal{N}}}_u^{\mathrm{\prime }}$, where ${\mathcal{N}}_u^{\mathrm{\prime }}\psi =\underset{ϵ\to 0}{lim}{\left.\frac{\mathcal{N}(u+ϵ\psi )-\mathcal{N}\left(u\right)}{ϵ}=\frac{\mathrm{d}}{\mathrm{d}ϵ}\mathcal{N}(u+ϵ\psi )\right|}_{ϵ=0},$ and ${\tilde{\mathcal{N}}}_u^{\mathrm{\prime }}$ is the adjoint of ${\mathcal{N}}_u^{\mathrm{\prime }}$. For the KdV equation, $\mathcal{N}\left(u\right)=u_t+\eta u{u}_x+{\epsilon }^2{u}_{xxx}$ which does not satisfy the symmetry condition.

Additional information

Funding

This research work was currently supported by National Basic Research Program of China (2014 CB845906), the ITER-China Program (2014GB124005), National Natural Science Foundation of China under (grant nos. 41274103, 11271195, 11321061, 11271357, 41504078).