Jump-keeping and Upwind Transfer in Multigrid for Upwind Schemes

Transfer operators in multigrid (MG) are generated, as a rule, on the basis of usual central formulas of weighting and linear interpolation to restriction and prolongation, respectively. This is quite naturally to elliptic problems with smooth coeecients, but not so evidently to hyperbolic equations in gasdynamics, where discontinuous solutions with shocks arise often. Nevertheless, beginning from Jameson's 1] and Ni's 2] pioneer papers, the central transfer operators are successively used in numerous Euler solvers. Probably this fact is explained due to including artiicial viscosity into schemes like 1], 2] to stabilise shocks. The terms of artiicial viscosity are represented by elliptic operators and they essentially innuence the convergence properties of a relaxation process. However, for upwind schemes where there is no artiicial viscosity, the MG procedure with standard prolongation and restriction operators can fail. Koren and Hemker 3], and Leclercq and Stouuet 4] suggest ways for improving transfer operators in MG by taking into account the upwindness of governing equations. Our own experience in generating and using upwind transfer operators has given also good results. Nevertheless, in our opinion, the question about what kind of transfer operators is needed for upwind schemes is still open. For instance, the authors of 3] point to upwind prolongation operator as the main reason of improving MG, whereas the authors of 4] improve MG only by using upwind restriction operator; we obtain a stable MG procedure with mixed central-upwind both restriction and prolongation operators, see Item 2. Note that central transfer operators have smoothing properties. Besides, all modiications are required only for ow cases with shocks. Therefore the main reason of failing standard MG is likely in the discontinuity of solutions. For instance, an appropriate modiication of transfer operators is used for elliptic problems with strongly discontinuous coeecients, see 5]. That is why we develop special transfer operators that treat functions with jumps. Such modiication turns out enough to obtain stable relaxation process for transonic ow calculations. 1. Jump-keeping transfer. We consider piecewise-continuous functions in d-dimensional unit parametric cube 0; 1] d. Let h and H = 2h be spacings of two nested uniform rectangular grids in our cube. We generate multidimensional transfer operators by the product of one-dimensional ones, i.e., at rst the given function is transferred for one parametric direction, then the obtained function is transferred for the next direction, and so on. Let index i run from 0 to I = …