Absorbing Boundary Condition for Nonlinear Hyperbolic Partial Diferential Equations with Unknown Riemann Invariants

Abstract

A general methodology for developing absorbing boundary conditions for general non-linear hyperbolic advective-diffusive equations with unknown Riemann invariants is presented. In problems where the Riemann invariants (RI) are known (e.g. the flow in a shallow rectangular channel, the gas flow equations), the imposition of non-reflective boundary conditions is straightforward. In problems where Riemann invariants are unknown (e.g. the flow in a non-rectangular channels, the stratified 2D shallow water flows) it is possible to impose that kind of conditions analyzing the projection of the Jacobians of advective flux functions onto normal directions to fictitious surfaces or boundaries. The advantage of the method is that it is very easy to implement in a finite element code and is only based on computing the advective flux functions (and the their Jacobian projections), then, imposing non-linear constraints via Lagrange Multipliers or Penalty Methods. The application of the dynamic absorbing boundary conditions to typical wave propagation problems with unknown Riemann invariants, like nonlinear Saint-Venant system of conservation laws for non-rectangular and non-prismatic 1D channels and stratified 1D/2D shallow water equations, is presented. Also, the new absorbent/dynamic condition can handle automatically the change of Jacobians structure when the flow regime changes from subcritical to supercritical and viceversa, or when recirculating zones are present in regions near fictitious walls.