Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

• Дата: 2015
• Автори: Levi, D., Martina, L., Winternitz, P.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2015
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/147153
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests / D. Levi, L. Martina, P. Winternitz // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 42 назв. — англ.

The main purpose of this article 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 Liouville equation are presented 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 one 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 significantly worse numerical results for solutions is presented and discussed.