On Multisymplecticity of Partitioned Runge-Kutta Methods

Previously, it has been shown that discretising a multi-Hamiltonian PDE in space and time with partitioned Runge–Kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law. However, these studies use the same partitioning of the variables into two partitions in both space and time. This gives rise to a large number of cases to be considered, each with its own set of conditions to be satisfied. We present here a much simpler set of conditions, covering all of these cases, where the variables are partitioned independently in space and time into an arbitrary number of partitions. In general, it is not known when such a discretisation of a multi-Hamiltonian PDE will give rise to a well defined numerical integrator. However, a numerical integrator that is explicit will typically be well defined. In this paper, we give sufficient conditions on a multi-Hamiltonian PDE for a Lobatto IIIA–IIIB discretisation in space to give rise to explicit ODEs and an algorithm for constructing these ODEs.