The variational approach to time discretization of Birkhoff's equations for infinite-dimensional systems

Abstract

Difference methods are widely used for the numerical solution of problems in mechanics and physics. When constructing discrete analogues, it is important to preserve the basic properties of the original differential problem. The main goal of this work is to discretize a system of equations of the form \(C(x,t,u)u_t + E(x,t,u_\alpha) = 0\), based on its functional — the Hamiltonian action. Necessary and sufficient conditions for potentiality with respect to a given bilinear form are obtained. The Hamiltonian action for this system is constructed and its representation in the form of Birkhoff's equations for infinite-dimensional systems is obtained. By approximating the constructed functional by its discrete analogue, a discrete-time analogue of Birkhoff's equations is obtained based on the variational principle. Theoretical results are illustrated by an example of a wave equation with axial symmetry.