A formally second‐order cell centred scheme for convection–diffusion equations on general grids

Abstract

SUMMARY We propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non‐conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second‐order spatial convergence rate for the Laplace equation on any unstructured two‐dimensional/three‐dimensional grids. As in the MUSCL method, the numerical convective fluxes are built with a prediction‐limitation process, which ensures that the discrete maximum principle is satisfied for pure convection problems. The limitation does not involve any geometrical reconstruction, thus allowing the use of completely general grids, in any space dimension. Copyright © 2012 John Wiley & Sons, Ltd.

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

“…For a first-order scheme, it consists in using an upwinding with respect to the material velocity only [27]; here, a higher order scheme is sought, and thus a MUSCL-like procedure is developed, see [41] for the original MUSCL scheme: we compute formally second-order in space fluxes and then apply a limiting procedure to obtain positivity under a CFL-like condition, since we use a explicit time discretization. Inspired from the work performed in [34] for the transport equation, this limiting step is purely algebraic: it does not require any geometric argument and thus works on quite general meshes. It is carefully designed to keep the pressure constant in the zones where it actually should be, and in particular across contact discontinuities.…”

“…As a consequence of this equation-per-equation process, the problem that we face is close to the program realized in [34], namely to build an approximation for a convection operator (satisfying a maximum principle) which is formally second order in space when the solution is regular, and preserves the range of variation of the unknowns even in case of shocks, by an adequate flux limiting procedure. The algorithm presented here is thus an extension of the scheme developped in [34]; in particular, contrary to most MUSCL reconstructions which use slope estimation and limiting, see e.g.…”

“…The algorithm presented here is thus an extension of the scheme developped in [34]; in particular, contrary to most MUSCL reconstructions which use slope estimation and limiting, see e.g. [3,39] for reviews and [31,5,8,9] for recent works, the limiting technique is here directly derived from stability conditions which are purely algebraic, in the sense that they do not require any geometric computation, and thus work with arbitrary meshes.…”

“…Compared to [34], the algorithm is however complicated by the requirement that the scheme should preserve pressure-constant zones, to avoid to destabilize the computation of contact discontinuities, or more precisely, of the one-dimensional contact discontinuity, across which the velocity is constant, the more difficult problem posed by slip interfaces in 2D or 3D being out of the scope of this study. In fine, this is realized by imposing to the face pressure (i.e.…”

“…For structured discretization, the value at the internal face σ = K|L is obtained as a weighted average of ρ K and ρ L . For a detailed description of the implemented interpolation algorithm in the general case, the reader is referred to [34].…”