Abstract
Symmetry preserving difference schemes approximating equations of Hamiltonian systems are presented in this paper. For holonomic systems in the Hamiltonian framework, the symmetrical operators are obtained by solving the determining equations of Lie symmetry with the Maple procedure. The difference type of symmetry preserving invariants are constructed based on the three points of the lattice and the characteristic equations. The difference scheme is constructed by using these discrete invariants. An example is presented to illustrate the applications of the results. The solutions of the invariant numerical schemes are compared to the noninvariant ones, the standard and the exact solutions.
Highlights
All variables are measured in a certain time interval, and all simulations can be implemented in meshes
Different types of difference equations which admit the different symmetry transformation operators can be obtained [10,11], and this theory is applied to search for discrete symmetries for dynamical systems [12,13]
The basic motivation for this research program is applying the method of symmetry preserving discretization to the holonomic Hamiltonian systems
Summary
All variables are measured in a certain time interval, and all simulations can be implemented in meshes. Different types of difference equations which admit the different symmetry transformation operators can be obtained [10,11], and this theory is applied to search for discrete symmetries for dynamical systems [12,13] Speaking, it needs to solve some linear partial finite-difference equations. Dorodnitsyn et al proposed approaches on the applications of Lie group theory to difference equations and discrete mechanical systems [19]. The goal of this article is to extend the method of invariant discretization of differential equations to the holonomic Hamiltonian systems This new method can preserve fundamental symmetric properties and improve the features of numerical algorithms in a Hamiltonian framework.
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