Alain Lerat, Christophe Corre and Gregoire Hanss
SINUMEF Lab. ENSAM
151 bd de l'Hopital, 75 013 Paris, France
email: lerat/corre/hanss@paris.ensam.fr
For multidimensional hyperbolic systems, we consider compact approximations that provide a high accuracy order not for each space derivatives treated apart as with classical Pade formulae, but with the whole residual r. For steady problems, r is the residual at steady state. For unsteady problems, r includes the time derivative but is still considered as a steady-state residual with respect to a dual time. This notion of residaul-based compactness avoids the solution of linear algebraic systems associated with usual compact schemes.
Dealing with the residual is also favourable for constructing the numerical dissipation. A residual-based dissipation contains derivatives of the residual only and thus is consistent with a partial differential operator vanishing at steady state. Such a dissipation can be first-order in the trabsient phase and therefore can provide robustness in a simple way, withut degrading the high accuracy at steady-state (in real or dual time) or the compactness.
High-order based schemes wiil be presented at the Workshop. Accuracy and efficiency will be discussed for multidimensional linear advection test-cases and for various compressible flow problems governed by the Euler or the Navier Stokes equation in steady and unsteady regimes.
References