definiteness of a scheme, or monotonicity and total variation diminishing constraints. These constraints are violated Many problems of fluid dynamics involve the coupled transport of several, density-like, dependent variables (for instance, densities by linear higher-order numerical approximations [11].
of mass and momenta in elastic flows). In this paper, a conservative Powerful methods nowadays allow for the implementaand synchronous flux-corrected transport (FCT) formalism is develtion of various constraints into numerical schemes. One oped which aims at a consistent transport of such variables. The method of wide applicability and of particular importance technique differs from traditional FCT algorithms in two respects.
to the present study is the flux-corrected transport (FCT)
First, the limiting of transportive fluxes of the primary variables (e.g., mass and momentum) does not derive from smooth estimates originated by and generalized by of the variables, but it derives from analytic constraints implied by Zalesak [36]. Succinct reviews of this technique can be the Lagrangian form of the governing continuity equations, which found in [31,19,35]. In geophysical applications, considerare imposed on the specific mixing ratios of the variables (e.g., ation is typically given to the transport problem velocity components). Second, the traditional FCT limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements. This procedure can
also be used in the framework of traditional FCT schemes, and a demonstration is provided that it can significantly reduce some of the pathological behaviors of FCT algorithms. Although the approach derived is applicable to the transport of arbitrary conserved
where v is an externally specified velocity vector and quantities, it is particularly useful for the synchronous transport of is an arbitrary density-like dependent variable. The FCT mass and momenta in elastic flows, where it assures intrinsic stabiltechnique makes comparative use of both a first-order and ity of the algorithm regardless of the magnitude of the mass-density a higher-order time-step, and it ensures in effect that new variable. This latter property becomes especially important in fluids with large density variations, or in models with a material ''vertical'' local extremes in the higher order integration of can only coordinate (e.g., geophysical hydrostatic stratified flows in develop if they do so already in the first-order time-step. isopycnic/isentropic coordinates), where material surfaces can col-This is a heuristic approach, but it is very successful in lapse to zero-mass layers admitting, therefore, arbitrarily large local providing smooth and nonlinearly stable integrations (cf.