In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrators are geometric integrators, meaning that they preserve the geometry of the problems; in particular, the numerical method preserves energy and momentum in some sense, similar to the partial differential equation itself. Examples of multisymplectic integrators include the Euler box scheme and the Preissman box scheme.
Multisymplectic equations A partial differential equation (PDE) is said to be a multisymplectic equation if it can be written in the form
K z t + L z x = ∇ S ( z ) , {\displaystyle Kz_{t}+Lz_{x}=\nabla S(z),}
where z ( t , x ) {\displaystyle z(t,x)} is the unknown, K {\displaystyle K} and L {\displaystyle L} are (constant) skew-symmetric matrices and ∇ S {\displaystyle \nabla S} denotes the gradient of S {\displaystyle S} . This is a natural generalization of J z t = ∇ H ( z ) {\displaystyle Jz_{t}=\nabla H(z)} , the form of a Hamiltonian ODE. Examples of multisymplectic PDEs include the nonlinear Klein–Gordon equation u t t − u x x = V ′ ( u ) {\displaystyle u_{tt}-u_{xx}=V'(u)} , or more generally the nonlinear wave equation u t t = ∂ x σ ′ ( u x ) − f ′ ( u ) {\displaystyle u_{tt}=\partial _{x}\sigma '(u_{x})-f'(u)} , and the KdV equation u t + u u x + u x x x = 0 {\displaystyle u_{t}+uu_{x}+u_{xxx}=0} . Define the 2-forms ω {\displaystyle \omega } and κ {\displaystyle \kappa } by
ω ( u , v ) = ⟨ K u , v ⟩ and κ ( u , v ) = ⟨ L u , v ⟩ {\displaystyle \omega (u,v)=\langle Ku,v\rangle \quad {\text{and}}\quad \kappa (u,v)=\langle Lu,v\rangle }
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle } denotes the dot product. The differential equation preserves symplecticity in the sense that
∂ t ω + ∂ x κ = 0. {\displaystyle \partial _{t}\omega +\partial _{x}\kappa =0.}
Taking the dot product of the PDE with u t {\displaystyle u_{t}} yields the local conservation law for energy:
∂ t E ( u ) + ∂ x F ( u ) = 0 {\displaystyle \partial _{t}E(u)+\partial _{x}F(u)=0}
where
E ( u ) = S ( u ) − 1 2 κ ( u x , u ) , F ( u ) = 1 2 κ ( u t , u ) . {\displaystyle {\begin{aligned}E(u)&=S(u)-{\tfrac {1}{2}}\kappa (u_{x},u),\\[1ex]F(u)&={\tfrac {1}{2}}\kappa (u_{t},u).\end{aligned}}}
The local conservation law for momentum is derived similarly:
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