Transformation groups in net spaces

Difference schemes with point symmetries and their numerical tests

“…In discrete descriptions, using difference equations, continuous symmetries are usually lost. It has been shown earlier [1,2,4,5,6,7,8,9] that it is possible to construct difference schemes that possess the same symmetries as their continuous limits. To achieve this, it is necessary to use difference schemes (equations and meshes) constructed out of the invariants of the corresponding Lie groups.…”

“…is of lower rank. The fact that difference schemes can be invariant under continuous Lie point transformations that act on the equations and on lattices was pointed out by Dorodnitsyn [2,4]. This approach has been used to classify and solve three-point difference schemes [5,6].…”

Geometric Foundations of Numerical Algorithms and Symmetry

“…(4.4). The moving frame construction has a significant advantage over the infinitesimal approach used by Dorodnitsyn, [10,11], in that it does not require the solution of partial differential equations in order to construct the multi-invariants. Given a G-invariant differential equation…”

“…Thus, the theory of multi-invariants is the theory of invariant numerical approximations! The basic idea of replacing differential invariants by joint invariants forms the foundation of Dorodnitsyn's approach to invariant numerical algorithms, [10,11], and also the invariant numerical approximations of differential invariant signatures in computer vision, [1,4,5,27].…”

“…The first instances of such schemes are the symplectic integrators arising in Hamiltonian mechanics, and the related energy conserving methods, [7,21,30]. The design of symmetry-based numerical approximation schemes for differential equations has been studied by various authors, including Shokin, [29], Dorodnitsyn, [10,11], Axford and Jaegers, [19], and Budd and Collins, [2]. These methods are closely related to the active area of geometric integration of ordinary differential equations on Lie groups, [3,22].…”