Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
This work provides a symmetry-preserving discretization for a specific PDE, which is incremental for researchers in geometric numerical integration.
The paper identifies the infinite-dimensional symmetry algebra of the real elliptic Liouville equation and presents an O(3,1)-invariant difference scheme that converges to the equation in polar coordinates, with a lattice invariant under O(2) rotations.
The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane $E_2$. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group $O(3,1)$ and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under $O(3,1)$ and is itself invariant under a subgroup of $O(3,1)$, namely the $O(2)$ rotations of the Euclidean plane.