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\title{Discretizing the Liouville equation}
\author{ \underline{Decio Levi}, Dipartimento di Matematica e Fisica, \\ Universit\`a Roma Tre \\Luigi Martina, Dipartimento di Matematica e Fisica, \\ Universit\`a del Salento \\ Pavel Winternitz, Dipartimento di Matematica e Fisica, \\ Universit\`a Roma Tre and CRM, Universit\'e de Montr\'eal}
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\maketitle
The main purpose of this presentation is to show how structure reflected in
partial differential equations can be preserved in a discrete world and
reflected in difference schemes.
The Liouville equation is the simplest periodic reduction of the integrable two dimensional Toda lattice (two continuous and one discrete variable) and is known to be linearizable by a transformation to the wave equation.
Three different structure preserving discretizations of the Liouville
equation are presented here and then used to solve specific boundary
value problems. The results are compared with exact solutions
satisfying the same boundary conditions. All three discretizations are
on four point lattices.
One preserves linearizability of the equation, another the infinite
dimensional symmetry group as higher symmetries, the third preserves the
maximal finite dimensional subgroup of the symmetry group as point
symmetries.
A 9-point invariant scheme that gives a better
approximation of the equation,
but worse numerical results for solutions is presented and discussed.
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