The design of conservative finite element discretisations for the vectorial modified KdV equation
For researchers simulating nonlinear wave phenomena, this method provides a numerically stable and accurate approach for long-time integration of the vectorial mKdV equation.
The authors designed a Galerkin scheme for the vectorial modified KdV equation that conserves energy up to machine precision, enabling accurate long-time simulations without numerical dissipation. Numerical experiments demonstrate asymptotic convergence and correct capture of soliton interactions.
We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. 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.