Spectral element schemes for high order partial differential equations : Application to the Korteweg-de Vries model

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that using the classical spectral element method, i.e. a high order continuous Galerkin approximation, it is possible to develop satisfactory schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants. The proposed approach is a priori easily extensible to other partial differential equations and to multidimensional problems.