We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg–de Vries equation with periodic boundary conditions. We demonstrate that the scheme conserves energy up to solver tolerance. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.
- Conservative finite element method
- Hamiltonian PDE
- Vectorial modified KdV equation
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics