High Order $C^0$-Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and 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 it is possible to develop reliable and effective schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) $H^1$-conformal Galerkin approximation, namely the Spectral Element Method. The proposed approach is {\it a priori} easily extensible to other partial differential equations and to multidimensional problems.