Francis Valiquette

Alexander Bihlo, Francis Valiquette

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.

Raphaël Rebelo, Francis Valiquette

The method of equivariant moving frames on multi-space is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical Burgers equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations on uniform rectangular meshes.

Alexander Bihlo, Francis Valiquette

Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.

Raphael Rebelo, Francis Valiquette

Given a differential equation with infinite-dimensional symmetry pseudo-group it is shown, using an example, that it is generally not possible to construct enough joint invariants to form an invariant numerical scheme of the equation. To circumvent this problem, we propose to discretize the symmetry pseudo-group action. Using the theory of moving frames, joint invariants of the discretized action are algorithmically constructed. Computer simulations indicate that numerical schemes constructed from these joint invariants can produce better numerical results than standard schemes.