CABARET is a general-purpose advection scheme which is suited for computational aeronautics and geophysics problems. For solving Navier-Stokes equations with Reynolds numbers of 104, the method gives a very good convergence without any additional preconditioning down to Mach numbers as low as 0.05-0.1. In particular for the MILES modelling of a hydrodynamic instability and free jet, a 2572 grid using CABARET is able to produce results comparable to a conventional second order method which would require at least 10252 grid points . Here, the CABARET method is 30 times more efficient.
For linear advection, the CABARET scheme is a modification of the non-dissipative and low-dispersive Second-order Upwind Leapfrog method . The modification consists of introducing separate conservation and flux variables that are staggered in space and time this results in a very compact, one cell in space and time computational stencil. In comparison to the standard finite-difference and finite-volume methods, in CABARET there is always an additional independent evolutionary variable, which gives the method the ability to preserve one more important property of the governing equations - the small phase and amplitude error. Traditionally conservation fluxes are computed at cell faces using cell-centre variable interpolation but with CABARET the conservation fluxes explicitly depend on the evolution of cell-centre and cell-face variables. For enforcing the non-oscillatory property of the solution, the CABARET scheme uses a low-dissipative non-linear flux correction that is directly based on the maximum principle for the flux variables.
The key role of the nonlinear correction in the MILES CABARET method is to remove the under-resolved fine scales from the solution so that an accurate balance between the numerical dissipation and dispersion errors is preserved. For the Navier-Stokes equations, each time iteration of the CABARET method consists of a conservation phase and a characteristic decomposition phase.
Phil Ridley 2011-02-01