Characteristics method for the formulation and computation of a free boundary cavitation problem

In this paper, a semidiscretized scheme based on characteristics method is analyzed when it is applied to a free boundary problem. The free boundary problem issued from a cavitation model in lubrication involves specific boundary conditions. The idea is to associate to the departure problem a sequence of variational inequalities depending on a discretization parameter. The theoretical convergence result from the solution of the semidiscretized problem to the solution of the continuous one is stated for a flux-imposed boundary condition. For this purpose, due to the specific boundary conditions, technical modifications of the classical version of characteristics are needed. Then, obstacle problem tools are applied to a set of variational inequalities with Neumann boundary conditions in order to obtain L∞(Ω) estimates which provide the convergence result. In this way, a result of existence of solution for the continuous problem and a theoretical justification for the usage of the proposed numerical methods are concluded. Finally, some numerical test examples are presented to illustrate the good performance of the method.