The Korteweg-de Vries equation and its symmetry-preserving discretization
For researchers in numerical analysis and mathematical physics, this work provides symmetry-preserving discretizations that maintain Galilean invariance, though the accuracy improvement is incremental.
The paper constructs invariant discretization schemes for the Korteweg-de Vries equation that preserve Galilean transformations and momentum, achieving accuracy comparable to standard schemes.
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.