Boundedness criteria

The main advantage of this scheme is that it satisfies the boundedness criteria unconditionally. A severe disadvantage associated with the UDS is that it is very numerically diffusive. 

The main advantage of this scheme is that it is not numerically diffusive. However, the scheme is dispersive and a severe disadvantage associated with the CDS is that it does not always satisfy the boundedness criteria. The CDS may become unstable when F /D < 2, because the east coefficient will be negative. This scheme is thus conditionally stable. For this reason, the CDS may produce oscillatory solutions like all approximations of order higher than one. 

Recently, the flux limiters (FL) and the normalized variables (NV) design principles that are employed to construct non-linear high resolution convection schemes have been shown to lead to the same constraints on the fluxes [107, 189, 206]. However, three different boundedness criteria are frequently employed, these are the total variational diminishing (TVD) [69], positivity [141] and the convection boundedness criterion (CBC) [58]. Thus, although these schemes have slightly different conceptual bases, the two approaches lead to mathematically equivalent schemes [189]. In multidisciplinary research fields, it happens now and then that mathematically equivalent schemes are invented independently more than once, often with different conceptual bases [189]. 

As we observed in Section IV it is frequently the case that the linearized approximation of, for example, Eq. (24), with/(t) = 0, exhibits an unstable equilibrium point x = 0 although one of the boundedness criteria in Section IV applies. In such a case a nontrivial solution x(t) of Eq. (24) will obviously be oscillatory. 

The situation is different where boundedness criteria and criteria implying stability for a finite range of initial perturbations are concerned. Progress is very likely to be made on hypotheses on K(t) implying boundedness of solutions of Eqs. (1) or (18). And, in all probability, these results will be of greater practical use than the criteria ensuring asymptotic stability in the large. 

BCGSTAB Bi-orthogonal Conjugate Gradient STABilized BFBC Bubbling Fluidized Bed Combustion CAFBC Circulating Atmospheric Fluidized Bed Combustor CARPT Computer Automated Radioactive Particle Tracer CBC Convection Boundedness Criterion... 

It can be seen that the central differencing scheme discussed above is conservative. The coefficients of CDS satisfy the Scarborough criterion. However, for uniform grid (Ae = 0.5), when the Peclet number is higher than 2, the coefficients oe will become negative [Fg > This violates the boundedness requirements and may... 

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