Abstract
When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.
Highlights
The development of the theory of heat transfer can be traced by publications and republications of books "Heat Transfer" by Isachenko V.P., Osipova V.A., Sukomel A.S., "Theory of heat and mass exchange" [1], "Theory of heat and mass transfer" by Lykov A.V., Mikhailov Yu.A. [2], "Computational heat transfer" by Samarsky A.A., Vabishchevich P.N. [3], and other publications in the subject area.In [4], the classification of methods for solving non-stationary nonlinear heat and mass transfer problems in solids was carried out
The above algorithm can be used to solve the elliptic equation with the introduction of fictitious time [13]. These complications, as noted above, are due to the inconsistency of the values sought from neighboring layers used in the finite difference equations
If we introduce the notation hy2 / hx2, the finite difference equations take the form uin, j 1
Summary
In [4], the classification of methods (variation, linearization, projection, etc.) for solving non-stationary nonlinear heat and mass transfer problems in solids was carried out. Traditional methods of variables separation, sources, Laplace, LaplaceCarson, Fourier, Hankel and other methods are used. The solutions to numerous heat transfer problems with different boundary conditions, including the mixed ones, are given in single and multilayer materials using the generalized variables. The thermal conductivity coefficients were obtained for various values of mass flow (5-58 kg/m2s) and heat flow (1-5.6 kW/m2). The effect of heat flow, mass flow and steam quality on the coefficient of thermal conductivity was analyzed. Based on the existing three-zone model, a new theoretical model has been developed for calculating the thermal conductivity coefficient of the bubble part in the mini-channel. With an absolute error of 9.72%, the new model provides a good agreement with experimental data
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