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Article information
2017 , Volume 22, ¹ 5, p.27-38
Vyatkin A.V., Kuchunova E.V., Shaidurov V.V.
Semi-Lagrangian method for two dimensional advection problem with discrete balance equation
In this paper, the semi-Lagrangian method (the generalized method of characteristics) is considered for the numerical solution of the two-dimensional (in space) continuity equation for the gas or fluid motion with given velocities. The geometric pattern of the method is represented by a fluid flow tube with an arbitrary two-dimensional rectangular cell at the upper time level constructed by passing flow lines (characteristics of the equation) backward in time to the intersection with the previous time level or the inflow surface. For the obtained three-dimensional (in the time-space coordinates) body, the Gauss -Ostrogradsky theorem is applied. Due to the absence of the gas flow across the side surfaces defined by trajectories, the relation between two-dimensional integrals of a solution at neighbouring time levels (in some cases on the inflow surface) is obtained. The paper primarily deals with the detailed approximation of these integrals resulting in explicit grid equations where the values of an approximate solution at the upper time level are expressed in terms of the values at the lower level and on the inflow surface. A fluid flow tube for gas issuing through a rectangular cell on the outflow surface is considered in a similar way. After the corresponding approximation, the quantity of issuing gas is expressed in terms of its integrals at the previous time level. The proposed methods of the approximation of integrals provide the fulfillment of the mass conservation law at the discrete level for the motion of gas between two time levels taking into account gas inflow and outflow. Numerical results confirming the fulfillment of the discrete conservation law are presented. Due to the dynamical adaptation of the grid pattern, the Courant -Friedrichs -Lewy condition for the time step size is removed.
[full text]
Keywords: semi-Lagrangian method, balance equation, advection problem
Author(s):
Vyatkin Alexander Vladimirovich
PhD.
Position: Scientific Secretary
Office: Institute of Computational Modeling of Siberian Branch of Russian Academy of Science
Address: 660036, Russia, Krasnoyarsk, Krasnoyarsk, Akademgorodok, 50/44
Phone Office: (391) 249-53-70
E-mail: vyatkin@icm.krasn.ru
SPIN-code: 4488-9118Kuchunova Elena Vladimirovna
PhD.
Position: Associate Professor
Office: Siberian Federal University
Address: 660041, Russia, Krasnoyarsk, 79 Svobodny pr.
Phone Office: (391) 206-20-87
E-mail: HKuchunova@sfu-kras.ru
SPIN-code: 9039-0784Shaidurov Vladimir Victorovich
Dr. , Correspondent member of RAS, Professor
Position: Head of Research
Office: Federal Research Center Krasnoyarsk Science Center of the Siberian Branch of the Russian Academy of Science
Address: 660036, Russia, Krasnoyarsk 36, Akademgorodok 50, building 44
Phone Office: (391) 243 27 56
E-mail: shaidurov04@gmail.com
SPIN-code: 7075-6423 References:
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Bibliography link:
Vyatkin A.V., Kuchunova E.V., Shaidurov V.V. Semi-Lagrangian method for two dimensional advection problem with discrete balance equation // Computational technologies. 2017. V. 22. ¹ 5. P. 27-38
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