A new local energy-preserving algorithm for the BBM equation

In this paper, a new local energy-preserving algorithm is proposed based on the temporal and spatial discretizations where the Average Vector Field (AVF) method and the implicit midpoint method are used for the temporal discretization and spatial discretization, respectively. In any local time-space region, the local mass and local energy are conserved by the algorithm. With periodic boundary conditions, it is worth noting that the global mass and global energy conservation law are also admitted. Numerical experiments are performed to support our theoretical analysis and show the conservation properties of the algorithm intuitively.