Difference scheme monotone

Some modification of the describing monotone difference scheme for divergent second-order equations was made by Golant (1978) and Ka-retkina (1980). In Andreev and Savin (1995) this scheme applies equally well to some singular-perturbed problems. Various classes of monotone difference schemes for elliptic equations of second order were composed by Samarskii and Vabishchevich (1995), Vabishchevich (1994) by means of the regularization principle with concern of difference schemes. 

Vabishchevich, P. (1994) Monotone difference schemes for convection-diffusion problems. Differential Equations, 30, 503-531 (in Russian). 

Accuracy of the difference scheme is 0(Af + Ar2), which could be reduced to 0(At2 4- Ar2) by means of the symmetrical difference scheme. In practice schemes with monotonously increasing spatial and temporal steps are usually used for these purposes [1, 9-11]. As r 1, Ar is small but increases with r whereas At increment is limited by the condition that the relative change of gm at any step should not exceed a given small value. Unlike the case of immobile particle reaction, the calculation of the functionals J[Z], (5.1.37) and (5.1.38), requires one-dimensional integration only which is not time-consuming. 

The numerical methods for solving equations like (8.2.17), (8.2.22) and (8.2.23) are discussed in Section 5.1. In practice the conservative difference schemes are widely used for solving differential equations with the accuracy of the order 0(At + Ar2) [21, 26, 27] used as well 0(Af2 4- Ar2) [25], Unlike mathematically similar equations for the A + B —> 0 reaction (Section 5.1), where the correlation functions vary monotonously in time, the... 

Lotka-Volterra model reveals different kind of autowave processes with the non-monotonous behaviour of the correlation functions accompanied by their great spatial gradients and rapid change in time. Due to this fact the space increment Ar time increment At was variable to ensure that the relative change of any variable in the kinetic equations does not exceed a given small value. The difference schemes described above were absolutely stable and a choice of coordinate and time mesh was controlled by additional calculations with reduced mesh. 

The convective terms were solved using a second order TVD scheme in space, and a first order explicit Euler scheme in time. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the smoothness monitor of van Leer [193] and the monotonic centered limiter [194]. The diffusive terms were discretized with a second order central difference scheme. The time-splitting scheme employed is of first order. 

Van Leer B (1974) Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J Comput Phys 14 361-370... 

Goncharov, A. L. Fryazinov, I. V. On the construction of monotone of difference schemes for the Navier-Stokes equations on devyatito of point patterns, Inst. Appl. Math. Keldysh RAN.M., 1986,93,14-16. 

The algorithm used for solving the problem of simulation of the simultaneous growth and dissolution of the oxygen and carbon precipitates due to the interaction of point defects during cooling of the crystal from the crystallization temperature is based on the monotonic explicit difference scheme of the first-order accuracy as applied to the Fokker-Planck equations. 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

It is worth mentioning here that the sign of r x) has had a significant impact on construction of monotone schemes. One way of providing a second-order approximation and taking care of this sign is connected with a monotone scheme with one-sided first difference derivatives for the equation with perturbed coefficients... 

There is one immediate problem with the m n classification scheme. It concerns the possibility of retrograde motion, i.e. the possibiUty that one of the angles 0 or 02, or both, do not increase monotonically while traversing the periodic orbit. In this case, two different periodic orbits may end up with the same m n label. It is currently not clear whether this situation can occur. A careful study has to be based on the equations of motion (10.3.3). 

It is further noticed that in the sense of preserving monotonicity of the solution a difference between the limiting processes of TVD and FCT schemes lies in that the TVD schemes usually are of one step, while the FCT is of two steps. FCT schemes are widely used for simulating time-dependent flows, but are less suited for steady-state calculations and therefore have had little influence in computational fluid dynamics applied to chemical reactor engineering. The TVD schemes, on the other hand, can be easily implemented into standard CFD codes by means of the deferred correction approach using flux limiters without enlarging the stencil. 

The main limitation of the weighted averaging scheme outlined above is that each intensive property varies monotonically with composition. Deviations from such behavior, which may occur because of specific interactions between different repeat units and produce a maximum or minimum in a property at an intermediate composition, are not taken into account. 

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