An improved finite-difference scheme for the conservation equations of matter

¹ TIIAME National Research University, Ecology and Water Resources Mamagement Department, 100000, 39, Kari Niyazov, Tashkent, Uzbekistan
² University of Texas at Austin, TX 78712, 110 Inner Campus Dr., Austin, USA

^* Corresponding author: pepiwm@gmail.com

Abstract

The finite-difference scheme of directed differences (the Courant-Isaacson-Ries scheme), which is widely used in the practice of aerohydrodynamic calculations, is studied theoretically and on the example of test problems. We applied the commonly used in practice Courant-Isakson-Ries directional difference scheme that allowed us to find and show distributions of velocities where the laws of the matter conservation are violated in the calculations in solving the matter conservation equations or the correspondence of the obtained solutions to the most general practical understandings on the essence of the matter transfer. A scheme free from the shortcomings of the Courant-Isaacson-Ries scheme has been constructed, tested, and proposed for use in aerohydrodynamic calculations by the finite difference method. Moreover, all the valuable properties of this well-known scheme are preserved. Among the maintained properties: are transportability, conservatism, stability in calculations, invariance, and adequacy of the essence of the physical phenomenon of the transfer of matter in space. The disadvantages of the new finite-difference scheme proposed for solving the equations of conservation of matter should be considered: an increase in the required RAM for storing electronic means of calculating information about the velocity field in memory and an increase in the number of calculations needed.