Partial Riemann problem, boundary conditions, and gas dynamics

TL;DR

This paper introduces the concept of partial Riemann problems as a generalization of classical Riemann problems, providing a new framework for boundary conditions in gas dynamics and conservation laws, with theoretical analysis and practical finite volume method implementation.

Contribution

It presents the notion of partial Riemann problems, extends the classical theory to general systems, and offers a practical finite volume method for boundary condition implementation.

Findings

Partial Riemann problem admits solutions in nonlinear wave classes.

Weak formulation of Dirichlet boundary conditions is established.

Practical finite volume method implementation demonstrated.

Abstract

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.