Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations
There exists a well known equivalence in Hamiltonian mechanics between the \(2n\) canonical equations of motion and the first-order partial differential Hamilton-Jacobi equation. In fact, when solving the Hamilton-Jacobi equation one is also obtaining a solution of the corresponding mechanical problem. This equivalence can be traced to the fact that both the Hamilton-Jacobi equation and the Hamiltonian equations of motion originate from a common variational principle. A solution of the Hamilton-Jacobi equation provides a generating function of a symplectic map. This connection has been exploited, in particular, from a numerical point of view to construct symplectic integrators for Hamiltonian systems of ordinary differential equations (ODEs) from generating functions. When considering the class of Hamiltonian partial differential equations (PDEs), one possible formulation consists in expressing the system as \[ M z_t + K z_x = \nabla_z H(z), \tag{1} \] where \(z \in \mathbb{R}^d\) is the state variable, \(M\) and \(K\) are \(d \times d\) skew-symmetric matrices, \(H\) is a smooth scalar function and \(\nabla_z\) represents the standard gradient operator. This formulation implies the existence of certain conservation laws, related to the symplectic structures both in space and time of equation (1). Numerical variants of Runge-Kutta (RK) and partitioned Runge-Kutta (PRK) methods can be designed that preserve a discrete version of these conservation laws. Such schemes are called multi-symplectic and, loosely speaking, they proceed by applying symplectic discretization procedures both in space and in time. The aim of the paper under review is to generalise the treatment carried out previously for Hamiltonian systems of ODEs to Hamiltonian PDEs in the multi-symplectic formulation. In particular, it is shown that multi-symplectic Hamiltonian systems can be considered as the characteristic equations of the so-called De Donder-Weyl (DW) Hamilton-Jacobi equation and its conservation laws can be related to generalised symmetries of this PDE. Based on this analysis, it is possible to construct explicitly generating functions for multi-symplectic RK and PRK methods.
scientific article; zbMATH DE number 5376033
scientific article