A method is proposed for applying the method of lines to solve two-dimensional hydrodynamic problems, which is an approximate analytical method for solving linear equations of mathematical physics and is developed in detail for the Dirichlet problem. In computational hydrodynamics, the use of the current and vorticity functions in the two-dimensional Navier-Stokes equations is often used, as a result of which the equations of elliptic and parabolic types are formed without pressure. Cross-differentiation compiles an equation for the hydrostatic pressure of the elliptical type with boundary conditions of the second kind. When applying the method of straight lines, the boundary conditions and equations are approximated with a uniform step along each of the Cartesian coordinates, and the resulting finite difference equations are represented as matrix equations for the unknowns defined for particular straight lines parallel to the coordinate axes. The structure of the basic three-diagonal transition matrices to finite-difference equations remains the same for the case of using separate equations for the current function, vorticity, and pressure. To solve the matrix equations, we use eigenvalues and eigenvectors of the main transition matrix, the values of whose elements are calculated only once. The exact solution of the equations is obtained with respect to a certain linear combination. The inverse transition to the desired functions is carried out using the fundamental matrices composed of the eigenvectors of the three-diagonal transition matrices to the finite difference equations. The method of straight lines is applied to solve equations for the current function and vorticity in the framework of the work. The boundary conditions and the right-hand sides of a finite difference equation with respect to one of the unknown functions are represented with the participation of another function. In this connection, a consistent approximation is formed. As an example of the application of the method, we chose the classical problem of the motion of an incompressible viscous fluid in a square domain, one boundary of which has a constant speed and the other boundaries are fixed. The proposed method has proven itself well in solving the equation of the current function, since by excluding the step of matching the results of applying the sweep method along different coordinate axes (i.e., no need to introduce a fictitious time or a sequential approximation method), the calculation time can be greatly reduced. The method was also used to solve the vorticity equation. In the case of discontinuous solutions, we were able to obtain convincing results of the problem only with small numbers of the Reynolds criterion. The results of the computational experiment showed that the use of the method of straight lines makes it possible to increase the accuracy of solving multidimensional linear equations of mathematical physics and reduces the calculation time significantly.
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Makhkamov, M. K. and Khujayev, J. I.
"APPLICATION OF THE METHOD OF LINES TO SOLVE PROBLEMS OF THE FLOW OF AN INCOMPRESSIBLE FLUID IN A RECTANGULAR REGION AT SMALL REYNOLDS NUMBERS,"
Scientific Bulletin. Physical and Mathematical Research: Vol. 1
, Article 13.
Available at: https://uzjournals.edu.uz/adu/vol1/iss1/13