Symmetry-Preserving Numerical Schemes

Abstract

In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.

Variational integrator for the rotating shallow‐water equations on the sphere

“…Important Lie groups for this paper are the group Diff(M ) of all smooth diffeomorphisms of the manifold M and its discrete version D(M) defined in (7). The Lie algebra of Diff(M ) is the space of vector fields u on M , endowed with (minus) the Lie bracket of vector fields [u, v] = u · ∇v − v · ∇u.…”

Symmetry-Preserving Finite Element Schemes: An Introductory Investigation

“…Therefore, when seeking to construct a symmetry-preserving numerical scheme for a particular differential equation, one can either start with the original strong form or work with a suitable weak form. The strong form of a differential equation is the starting point for constructing symmetry-preserving finite difference schemes, which is the route that has been taken so far in the literature, [1][2][3][4][5][6][7]10,11,[13][14][15][16][17]21,23,24,26,32,35]. On the other hand, the weak form is the starting point for constructing symmetry-preserving finite element schemes, which is the focus of the present paper.…”

“…See [7] for further details. In other words, the Lie group G induces an action on each fiber of J [2] via the product action.…”

“…In the following, we consider the semi-discretization of Burgers' equation where the spatial variable x is discretized and the time variable t remains a continuous variable. In this setting, the interpolating coefficients in the approximation (7) of the solution now become functions of t:…”

Construction of invariant compact finite-difference schemes

“…In the first group [20][21][22][23][24], invariants of difference equations are determined through Lie's infinitesimal approach, and then, a set of these invariants are used to construct invariant schemes that converge to the original differential equations in the continuous limit. In the other group [33][34][35][36][37][38][39][40][41][42][43], point transformations based on symmetry groups of differential equations are applied to some base (non-invariant) numerical schemes, and the unknown symmetry parameters of these transformations are determined through moving frames that are based on Cartan's method of normalization [44].…”