A method of coupling non-linear hyperbolic systems: Examples in CFD and plasma physics
5In the modelling of complex problems, different mathematical models are frequently used in different regions of interest. For instance, one can take into account some physical effects (multi-dimensional effects, compressibility, temperature variations, Joule effect in the following example, etc.) in some domain where they are supposed to be important or assume that they are negligible elsewhere which amounts to drop the corresponding terms in the equations of the complete model. We have studied an example of this situation in the context of plasma fluids with two different current densities. Also different closure relations may be considered. A simple example of this last situation which we analyse below is provided by the flow of two perfect fluids with different equations of state separated by a moving interface. This leads to couple different systems and we consider a coupling procedure which was initially motivated by numerical considerations (see References [1,2]). We are thus interested in the coupling of non-linear hyperbolic systems of conservation laws at an interface which we assume, for simplicity, is fixed. In the scalar case, the complete study from both mathematical and numerical points of view was considered in Reference [3]. Here, we take up the coupling of systems and the situation is far more complicated although we restrict ourselves to the one-dimensional case and to systems of the same size. Only in the linear case is it possible to justify completely the analysis of the continuous problem. We can also study coupled Riemann problem corresponding to an initial data constant on each side of the interface. Though simple in their formulation, they already provide interesting features.
conservation laws boundary value problems numerical methods