Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 052, 23 pages      arXiv:1209.5028      https://doi.org/10.3842/SIGMA.2013.052
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Invariant Discretization Schemes Using Evolution-Projection Techniques

Alexander Bihlo a, b and Jean-Christophe Nave b
a) Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada
b) Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montréal (QC) H3A 2K6, Canada

Received September 27, 2012, in final form July 28, 2013; Published online August 01, 2013

Abstract
Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.

Key words: invariant numerical schemes; moving frame; evolution-projection method; heat equation.

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