Invariantization of numerical schemes using moving frames

Abstract

This paper deals with a geometric technique to construct numerical schemes for differential equations that inherit Lie symmetries. The moving frame method enables one to adjust existing numerical schemes in a geometric manner and systematically construct proper invariant versions of them. Invariantization works as an adaptive transformation on numerical solutions, improving their accuracy greatly. Error reduction in the Runge-Kutta method by invariantization is studied through several applications including a harmonic oscillator and a Hamiltonian system. (2000): 65L12, 70G65. AMS subject classification

A review of some geometric integrators

“…However, combining these invariants into a numerically stable scheme is rather complicated. Instead, we follow the idea in [79][80][81], which consists in modifying classical schemes so as to make them invariant under the symmetry group. To show how to do this, we need to formalize the notion of a discretization.…”

“…It is then important that numerical schemes do not break symmetries if one wishes to reproduce numerically the cited model solutions and properties. Preserving the symmetries of the equations at the discrete level is the founding key of invariant integrators [77][78][79][80][81][82][83], as will be seen in "Invariant integrators" section. Note that the symmetry structure does not exclude simplecticity structure.…”

“…Invariant integrators are dealt with in "Invariant integrators" section. The approach used is that of Kim [79,133] and Chhay and Hamdouni [80] which consists in modifying classical schemes to make them invariant. At last, discrete exterior calculus is described in "Discrete exterior calculus" section.…”

Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation

“…But all of the schemes are not competitive, nor at least as effective as the original classical scheme. The first applications of invariantization have been presented by Kim for ordinary differential equations (ODE) and some partial differential equations to convince that this process can lead to invariant schemes that outperform classical schemes [32,33]. But the way to determine relevant moving frames is not well established yet [34].…”

Recent Advances in the Theory and Application of Lie Pseudo-Groups

“…Recent advances began with [27], that proposed a new approach to the classical theory of moving frames for general transformation groups. In the case of finite-dimensional Lie group actions, the reformulation of a moving frame, [14,30], as an equivariant map back to the Lie group, [28], proved to be amazingly powerful, sparking a host of new tools, new results, and new applications, including complete classifications of differential invariants and their syzygies, [31,69,71], equivalence, symmetry, and rigidity properties of submanifolds, [28], computation of symmetry groups and classification of partial differential equations, [49,59], invariant signatures in computer vision, [4,8,12,67], joint invariants and joint differential invariants [9,67], rational and algebraic invariants of algebraic group actions [32,33], invariant numerical algorithms [38,68,95], classical invariant theory [5,66], Poisson geometry and solitons [50,51,52], the calculus of variations and geometric flows, [39,70], invariants and covariants of Killing tensors, with applications to general relativity, separation of variables, and Hamiltonian systems, [56,57], and invariants of Lie algebras with applications in quantum mechanics, [10]. Subsequently, building on the examples presented in [27], a comparable moving frame theory for general Lie pseudo-group actions was established, [72,73,74], and applied to several significant examples, [19,…”