On the strong monotonicity of the CABARET scheme

Abstract

The strong monotonicity of the CABARET scheme with single flux correction is analyzed as applied to the linear advection equation. It is shown that the scheme is strongly monotone (has the NED property) at Courant numbers , for which it is monotone. Test computations illus trating this property of the CABARET scheme are presented.

On the construction of compact difference schemes

“…As a result, a scheme satisfying it transmits the Rankine-Hugoniot ε conditions [13] through neighborhoods of the dis continuity lines of the exact solution to higher accu racy.…”

“…The concept of weak approx imation for a difference scheme as applied to a hyper bolic system of conservation laws was introduced in [12], where criteria for such approximation, including to higher order accuracy, were obtained. For stable schemes, it was shown in [10] that the higher order of accuracy of weak approximations guarantees that the Rankine-Hugoniot conditions are transmitted across smeared shock fronts to higher accuracy and, as a result, a higher convergence rate is observed in areas of their influence.…”

“…It was shown in [10] that, in contrast to explicit two level (in time) conservative schemes [11], sym metric compact schemes approximating hyperbolic systems of conservation laws have the important prop erty that their classical approximations for smooth solutions are as accurate as weak approximations for discontinuous solutions. The concept of weak approx imation for a difference scheme as applied to a hyper bolic system of conservation laws was introduced in [12], where criteria for such approximation, including to higher order accuracy, were obtained.…”

“…This means that has the form (15) For the coefficients a 0j and a 1j of the operators and , which approximate y 0 (x) and (x), we com bine the last equation in (7) at l = γ = 1, the first equa tion in (9), and system (10) at n = 3 and r = 1 and take into account (14) to obtain the closed system…”

Modeling process of embolization arteriovenous malformation on the basis of two-phase filtration model

“…For this reason, the scheme possesses unique dissipative and dispersive properties [8]. In view of the special flux correction and certain constrains during approximation of initial data, the CABARET scheme is monotonic and strongly monotonic [9] at Courant numbers r ≤ 0.5. Therefore, the value r < 0.5 is used in the computations described below.…”