Local structure-preserving algorithms for the “good” Boussinesq equation

In this paper, we derive a series of local structure-preserving algorithms for the “good” Boussinesq equation, including multisymplectic geometric structure-preserving algorithms, local energy-preserving algorithms and local momentum-preserving algorithms. The outstanding advantage of the proposed algorithms is that they conserve these local structures in any time-space region exactly. For example, the proposed local energy-preserving algorithms preserve the local energy conservation law in any local domain. Therefore, the local structure-preserving algorithms overcome the shortage of global structure-preserving algorithms on the boundary conditions. Especially, with suitable boundary conditions such as periodic or homogeneous boundary conditions, the local structure-preserving algorithms will be global structure-preserving algorithms. Numerical results verify the theoretical analysis.