韩国裸舞

听

Seminar Physique Math茅matique

Discretizing the Liouville equation preserving the symmetries

The main purpose of this seminar is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the hyperbolic Liouville equation are presented and then used to solve a specific boundary value problem. The results are compared with the 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 one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme is briefly discussed at the end as well as the extension to the elliptic Liouville equation.