An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations

A Galerkin scheme is presented for a class of conservative nonlinear dispersive equations, such as the Camassa–Holm equation and the regularized long wave equation. The scheme has two advantageous features: first, it is conservative in that it keeps the discrete analogue of the continuous energy conservation property in the original equations; second, it can be formulated only with cheap H¹$H^1$-elements even if the original equations include third derivative u_xxx$u_{\mathit{xxx}}$. Numerical experiments confirm the stability and effectiveness of the proposed scheme.