Backward Error Analysis for a Multi-Symplectic Integrator

In this note we provide a backward error analysis of the numerical solution of a multi-symplectic (MS) Hamiltonian PDE of the form (1)$$ Mz_t + Kz_x = \nabla _z S(z),\quad z \in R^n $$ where M, K ∈ Rn×n are skew-symmetric matrices and S(z) is a smooth function of z(x, t) (the vector of state variables). The variational equation associated with (1) is is given by (2)$$ {\text{Mdz}}_{\text{t}} {\text{ + Kdz}}_{\text{x}} {\text{ = D}}_{{\text{zz}}} {\text{S(z)dz }} $$ Taking the wedge product of (2) with dz, and letting ω = 1/2 dz Λ M dz and κ = 1/2 dz Λ K dz, we obtain the MS conservation law (3)$$ \omega _t + \kappa _x = 0 $$ The two forms ω and κ define a symplectic structure associated with time and space, respectively. An important consequence of the MS structure is that when the Hamiltonian S(z) is independent of t and x, the PDE has local energy and momentum conservation laws [1, 2] (4)$$ E_t + F_x = 0,\quad E = S(z) + z^T Kz_x ,\quad F = - z_t ^TKz $$(5)$$ I_t + G_x = 0,\quad G = S(z) + z^T Mz_t ,\quad I = - z_x^T Mz $$ For periodic boundary conditions, the local conservation laws can be integrated in x to obtain global conservation of energy and momentum.