Application of Lie symmetries to difference equations and boundary value problems

Abstract

Lie groups of point transformations are applied to nonlinear hyperbolic partial differential equations in particular the sine-Gordon equation. Point symmetries of finite difference scheme for the sine-Gordon equation are obtained via methods developed from the existing literature. The method is extended to nonlinear partial difference equations and based on an algorithm that determines infinitesimal generators of the equation. Symmetries that leave the equation and the mesh invariant simultaneously is presented. It is shown that the sine-Gordon equation conserves the entire symmetry of the original differential form in its finite-difference model. Boundary value problems for differential and difference equations are also considered and the invariance of their boundary curves and boundary conditions under the Lie point symmetries of the associated equations is analyzed in both differential and difference forms. Symmetries affect the equations, mesh, boundaries and boundary conditions at the same time. The invariant discretization of the difference problem corresponding to boundary value problem for the sine-Gordon equation is studied.