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.
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 ,
,
, with
,
. This pseudo-group is considered in Example 3.20.
3. It is customary to use the notation to denote the value of the function
at the point
, and this is the convention used in Sections 3–5. In Equation (1.8), the subscript attached to the diffeomorphism
has a different meaning. Here, the subscript
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).