The Korteweg-de Vries equation and its symmetry-preserving discretization

Abstract

The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving Galilean transformations which is demonstrated numerically as well.

24 pages, 3 figures, 5 tables

References in corpus (4)

• Difference schemes with point symmetries and their numerical tests
• Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
• First integrals of ordinary difference equations beyond Lagrangian methods
• First integrals of ordinary difference equations which do not possess a variational formulation

Cited by in corpus (3)

• Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization