Applications of Group Theory in Computation—A Survey
Nail H. Ibragimov, W.F. Ames, R. L. Anderson, В. А. Дородницын, E. V. Ferapontov, Rafail Kavyevich Gazizov, Nail H. Ibragimov, Sergey R. Svirshchevskii · 2023
During the past three decades, research on and application of the group properties of differential equations has generated a substantial literature. Because of the versatility of Lie&s;s ideas, it is not surprising that group analysis would also find applications in numerical analysis. The passage from a partial differential equation to a difference scheme inhibits a group analysis because the difference operators do not usually have the same group properties as the differential operators. A related approach to that of Janenko and Shokin has been introduced by Postell [1990] and Ames, Postell, and Adams [1993], In addition to classical optimal finite-difference schemes, they introduce invariant algorithms in which the terms added to a basic finite-difference scheme are selected to make the method high order and also to preserve some or all of the original equation&s;s symmetry group. Examples include both hyperbolic and parabolic partial differential equations.