Section: New Results

Spectral element schemes for dispersive equations

Participants : Sebastian Minjeaud, Richard Pasquetti.

S. Minjeaud and R. Pasquetti have addressed the Korteweg-de Vries equation as an interesting model of high order PDE, in order to 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 (SEM). The proposed approach relies on the introduction of additional variables that can be trivially eliminated, because the SEM mass matrix is diagonal, thus allowing to define discrete high order differentiation operators. Highly accurate RK IMEX schemes are used in time, with implicit treatment of the third order term and explicit treatment of the convective one. While the conservation of the mass invariant is natural, the conservation of the energy invariant is enforced by interpolation between embedded IMEX schemes, with preservation of the time discretization accuracy. Applications to several test problems have shown the robustness and accuracy of the proposed method, that is a priori easily extensible to other PDEs and to multidimensional problems (See [38]).