Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation
The authors investigate the solutions of the well-known KdV equation in the case of periodic boundary conditions. To show the efficiency of the solutions, the authors apply a Fourier-Galerkin semidiscretization in space. This yields in a first step a Hamiltonian ODE problem. Next, in a second step, by observing an energy conservation in the Hamiltonian boundary value problem, the authors develop solutions in time by means of an energy-conserving method evolving Sylvester-type equations. The efficiency of the method is improved by the implementation of several numerical examples. \par Overall, it is an interesting work as it considers a famous problem of understanding the KdV equation which yields automatically an advance in understanding related problems such as wave propagation on the surface of shallow water, solitons, etc. \par The list of references is somehow exhaustive and reflects a good literature on the problem and the good background of the authors. \par I think finally that this work may open important directions by considering it on more complicated spaces such as fractal domains where a suitable notion of differentiablility, energy, periodicity should be defined.
Hamiltonian partial differential equations
scientific article; zbMATH DE number 7007561
scientific article