Partial Riemann problem, boundary conditions, and gas dynamics
The work provides a theoretical framework for handling boundary conditions in nonlinear conservation laws, which is relevant for computational fluid dynamics but is primarily a mathematical extension.
This paper introduces the concept of the partial Riemann problem, a generalization of the classical Riemann problem that incorporates boundary manifolds. It proves existence of solutions for general conservation laws and demonstrates applications to gas dynamics with finite volume methods.
We introduce in this contribution the notion of partial Riemann problem. Recall that the Riemann problem describes a shock tube interaction between two given states ; the partial Riemann problem is a generalization of the previous concept and introduces the notion of boundary manifold. In what follows, we first recall very classical notions concerning gas dynamics and the associated Riemann problem. In a second part, we introduce the partial Riemann problem for general systems of conservation laws and proves that this problem admits a solution in some class of appropriate nonlinear waves. In section 3, we recall the linearized analysis with the method of characteristics, introduce the weak formulation of the Dirichlet boundary condition for nonlinear situations in terms of the partial Riemann problem and show that lot of physically relevant situations are described with this theoretical framework. In the last paragraph, we propose a practical implementation of the previous onsiderations with the finite volume method.