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.
Contents
Multisymplectic equations
A partial differential equation (PDE) is said to be a multisymplectic equation if it can be written in the form
where
Examples of multisymplectic PDEs include the nonlinear Klein–Gordon equation
Define the 2-forms
where
Taking the dot product of the PDE with
The local conservation law for momentum is derived similarly:
The Euler box scheme
A multisymplectic integrator is a numerical method for solving multisymplectic PDEs whose numerical solution conserves a discrete form of symplecticity. One example is the Euler box scheme, which is derived by applying the symplectic Euler method to each independent variable.
The Euler box scheme uses a splitting of the skewsymmetric matrices
For instance, one can take
Now introduce a uniform grid and let
where the finite difference operators are defined by
The Euler box scheme is a first-order method, which satisfies the discrete conservation law
Preissman box scheme
Another multisymplectic integrator is the Preissman box scheme, which was introduced by Preissman in the context of hyperbolic PDEs. It is also known as the centred cell scheme. The Preissman box scheme can be derived by applying the Implicit midpoint rule, which is a symplectic integrator, to each of the independent variables. This leads to the scheme
where the finite difference operators
The Preissman box scheme is a second-order multisymplectic integrator which satisfies the discrete conservation law