Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups

Abstract

This paper is concerned with the invariant discretization of differential equations admitting infinite-dimensional symmetry groups. By way of example, we first show that there are differential equations with infinite-dimensional symmetry groups that do not admit enough joint invariants preventing the construction of invariant finite difference approximations. To solve this shortage of joint invariants we propose to discretize the pseudo-group action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudo-group can produce better numerical results than standard schemes.

Keywords::

• infinite-dimensional Lie pseudo-groups
• joint invariants
• moving frames

MSC (2010) Classification::

• 58J70
• 65N06

Acknowledgements

We thank Alexander Bihlo for stimulating discussions on the project, and Pavel Winternitz for his comments on the manuscript. The research of Raphaël Rebelo was supported in part by an FQRNT Doctoral Research Scholarship while the research of Francis Valiquette was supported in part by an AARMS Postdoctoral Fellowship.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

^1. Email: [email protected]

^2. Equation (1.3) admits a larger symmetry group given by $X=f\left(x\right)$, $Y=g\left(y\right)$, $U=u/(f\prime (x) g\prime (y\left)\right)$, with $f$, $g\in 𝒟\left(\mathbb{R}\right)$. This pseudo-group is considered in Example 3.20.

^3. It is customary to use the notation $f_{m,n}=f\left(x_{m,n}\right)$ to denote the value of the function $f\left(x\right)$ at the point $x_{m,n}$, and this is the convention used in Sections 3–5. In Equation (1.8), the subscript attached to the diffeomorphism $f_{m,n}\left(x_{m,n}\right)$ has a different meaning. Here, the subscript $(m,n)$ is used to denote different diffeomorphisms. Thus, the pseudo-group (1.5) is contained in the Lie completion (1.8). This particular use of the subscript only occurs in (1.8).