Weak Implementation of Boundary Conditions for the Finite-Volume Method

Abstract: The Euler equations consist of conservation laws and describe a fluid in motion without viscous forces and heat conduction. They are non-linear and the solutions are often discontinuous. Therefore proofs of convergence are hard to give and the existing results are lacking. A new concept are weakly imposed characteristic boundary conditions, where a priori given boundary data only enter the scheme via ingoing characteristics. Thus a numerical solution for the boundary points is obtained as well. In combination with the node-centred finite-volume method, approximations of two-dimensional steady conservation laws can be made stable. Weakly imposed characteristic boundary conditions are compared to weakly imposed prescribed fluxes for the steady Euler equations, where the residuals converged to 10^-11 and 10^-3, respectively. The performance of these boundary conditions is investigated further for different grid sizes and the unsteady Euler equations.