A modified Lagrange---Galerkin method for a fluid-rigid system with discontinuous density

In this paper, we propose a new characteristics method for the discretization of the two dimensional fluid-rigid body problem in the case where the densities of the fluid and the solid are different. The method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. To take into account the material derivative, we construct a special characteristic function which maps the approximate rigid body at the (k + 1)-th discrete time level into the approximate rigid body at k-th time. Convergence results are proved for both semi-discrete and fully-discrete schemes.