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Article information
2018 , Volume 23, ¹ 2, p.37-54
Kovyrkina O.A., Ostapenko V.V.
On monotonicity and accuracy of CABARET scheme for calculation of weak solutions with shocks
We studied the monotonicity and the accuracy of the modified CABARET scheme approximating the quasilinear hyperbolic system of conservation laws. We obtained conditions under which this scheme preserves the monotonicity of the finite-difference solution with respect to invariants of the linear approximation of the considering system. As a specific example, we considered the approximation for the system of conservation laws for shallow water theory. To determine the accuracy of the scheme in the regions influenced by shocks that propagate with variable velocity and behind the fronts of which a non-constant solution is formed, series of test calculations on the sequence of contracting grids were carried out. These calculations enabled us to apply the Runge rule to define the order of convergence of the finite-difference solution. To estimate the accuracy of the modified CABARET scheme, the orders of its integral and local convergence are calculated, as well as local disbalances in calculating of the absolute value of averaged basis functions vector and local disbalances in calculating of the absolute value of the components of the averaged vector of invariants. The order of the integral convergence makes it possible to estimate the accuracy of finite-difference schemes in translation of the RankineHugoniot conditions across a non-stationary shock wave front. In this case, the order of integral convergence is calculated for the finite-difference solution itself (rather than for its absolute value, as in the L1 norm) on spatial intervals crossing the shock wave. It is shown that, like TVD-schemes of high order approximation on smooth solutions, the CABARET scheme being formally of the second order, in spite of high accuracy in the localization of shocks, reduces the order of integral convergence to the first order on the integration intervals, one of the boundaries of which is in the region influenced by a shock. The main reason of this decrease in accuracy is that the flux correction used in the CABARET scheme to its monotonization leads to the decrease in the smoothness of the finite-difference fluxes, which in turn leads to the decrease in the order for approximation of Rankine-Hugoniot conditions on the shocks. As a result, the local accuracy of the CABARET scheme in the regions of shocks influence also decreases to about the first order.
[full text]
Keywords: hyperbolic system of conservation laws, monotonicity and accuracy of CABARET scheme, shallow water flows, shocks
doi: 10.25743/ICT.2018.23.12757
Author(s):
Kovyrkina Olyana Aleksandrovna
PhD.
Position: Senior Research Scientist
Office: Lavrentyev Institute of Hydrodynamics SB RAS
Address: 630090, Russia, Novosibirsk, 15 Lavrentyev pr.
Phone Office: (383)333-22-01
E-mail: olyana@ngs.ru
Ostapenko Vladimir Viktorovich
Dr. , Professor
Position: General Scientist
Office: Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the RAS
Address: 630090, Russia, Novosibirsk, Ac. Lavrentyev Ave., 15
Phone Office: (383)333-22-01
E-mail: ostigil@mail.ru
SPIN-code: 1676-5882 References:
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Bibliography link:
Kovyrkina O.A., Ostapenko V.V. On monotonicity and accuracy of CABARET scheme for calculation of weak solutions with shocks // Computational technologies. 2018. V. 23. ¹ 2. P. 37-54
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