Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations
zbMATH Open document ID: 1158.65085
author: Jialin Hong
author: Yajuan Sun
published in: Numerische Mathematik
publication date: 2008-12-01T00:00:00Z
review text: 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 cons…
reviewed by: Fernando Casas
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 70H20
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 5376033
zbMATH Keywords: multi-symplectic Runge-Kutta methods
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: generating function
zbMATH Keywords: Hamiltonian equations of motion
zbMATH Keywords: conservation laws
MaRDI profile type: Publication
full work available at URL: s00211-008-0170-x
OpenAlex ID: W2141124203
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DOI: 10.1007/S00211-008-0170-X
description: scientific article; zbMATH DE number 5376033