image: Comparing their data (the upper graph) with the data of colleagues (the lower graph), scientists concluded that their scheme has about 2 orders of magnitude (100 times) less error and a much slower increase in the error in time view more
Credit: Vladimir Gerdt
A group of researchers from Russia and Italy have conducted a study that produced a more precise scheme of numerical solution of incompressible Navier-Stokes equations for plane motion than that existed before. Details of the research can be found in the Applied Mathematics and Computation journal.
The Navier-Stokes equations are a system of differential equations that can be used to describe the motion of viscous Newtonian fluid. In many cases the dependence of the fluid density from coordinates and time can be neglected, and the density can be considered constant, making the fluid incompressible. With safe approximation, water is one of such fluids.
The Navier-Stokes equations are named after French physicist Claude-Louis Navier and British matematician George Gabriel Stokes. The problem of existence and smoothness of solutions to the Navier-Stokes equations is one of seven problems of the Millennium, the solution to which is rewarded with one million dollars by the Clay Mathematics Institute, USA.
The scheme built allows solving the equations for a flow of incompressible fluid efficiently, using numerical methods. Such fluids are used in different technological processes.
The calculations in the work are given for two-dimensional flow (by axes X and Y plus the time variable). This significantly simplifies the analysis of resulting difference scheme and the work itself, as there is an exact non-stationary solution for that case, which can be compared to the results obtained by approximate methods. Using computing experiments and comparisons with exact solutions, the researchers have verified the obtained numerical solution scheme qualitatively and quantitatively for laminar flow (i.e., one without mixing of the fluid layers and surges), leaving out turbulent flows, in which vortexes are formed and exact solutions are not possible.
"For example, when you turn on a faucet a little, the flow is slow. But when you turn it on fully, under high pressure the flow starts to whirl. This is turbulence, and it is hard to simulate numerically, - one of the authors, prof. Vladimir Gerdt, Doctor of Physical and Mathematical Sciences from Peoples' Friendship University of Russia (RUDN University), commented. - The solutions we study are rather smooth. Turbulent flows, in general, cannot be described with a high precision, and it requires supercomputer technology." To solve the equations with numerical methods, one needs to switch from differential equations to algebraic difference (discrete) equations. "Our goal is to build a good scheme which would with high precision solve a certain problem using numerical methods, as well as meet a number of special properties on top of numerical methods", - the mathematician told us.
According to him, in the future the researchers will be able to continue their work and turn to studying three-dimensional solutions, but first they need to make sure their method works well enough.
"When we specify the starting condition that satisfies the point solution, we look at how our numerical solution behaves depending from time, how much it will deviate from the exact solution. Other schemes that are known and studied in our paper have substantially larger errors that grow rather quickly with time, while our error is very small and grows very slowly, it is almost a constant, - the researcher told us. - We compared it to standard numerical methods by other authors and saw that our scheme is better. This is the main result of our work."
align="center">###Research was carried out in collaboration with researchers from Università di Bari (Italy), National Research Saratov State University (Saratov), and from Joint Institute for Nuclear Research (Dubna).
Journal
Applied Mathematics and Computation