Finite volume schemes for hyperbolic nonconservative systems
- Analysis of the general theory of hyperbolic nonconservative systems.
- Development of finite volume schemes for solving nonconservative problems: extension of the classical Roe scheme.
- Use of different kinds of reconstruction to build high-order finite volume schemes for hyperbolic nonconservative systems.
- Applications to conservation laws with source or coupling terms: oceanographic models.
Oceanographic models based on finite volumes
- Design of a unidimensional bi-layer model for the simulation of water exchange in the Strait of Gibraltar, based on finite volumes techniques. This model includes the variations of the bottom and the width, the nonlinear terms and the movement of the sea surface.
- Generalization of classic upwind schemes to a class of coupled systems of conservation laws with source term, that comprises, as a particular case, the equations of the bi-layer model designed. The solutions of the model have been validated by comparison with another aproximations found in the literature.
- Mathematical analysis of the equations of the two-layer model.
Oceanographic models based on finite elements
- Design of a bidimensional bi-layer model for the numerical simulation of water mass circulation in the Alboran sea, based on the finite elements method.
- Introduction of some variants of the Bermúdez-Moreno duality algorithm for the resolution of variational inequalities. This variants have proved to be of great efficiency in the resolution of the nonlinear problems arising in the model based on finite elements.
- Introduction of a technique for the computation of preconditioning matrices, well adapted for solving the linear problems arising when the algorithms in the preceding point are applied.
Ocean-atmosphere coupled models
- Design of a hybrid coupled model, resulting from the coupling between a general model of ocean circulation, that it is discretized and analyzed using finite differences, and a statistical model of the atmosphere.
- Application of the model to the study of the annual and interannual variability in the Tropical Pacific, this latter mainly represented by the "El Niño" phenomenon.
- Introduction of a novel technique for a posteriori error estimations, based on the use of metrics for the anisotropic adaptation of non-structured meshes.
- Design of software for mesh generation and adaptation.
- Improvements of the Farin's algorithm for mesh adaptation on surfaces, without changing of the corresponding geometries.
- Design of a conservative interpolation technique, useful for the mesh adaptation for the numerical resolution of conservations laws.