Physical and Metallurgical Faculty
Speciality: Thermal power plants
I'll review the literature on the difference schemes are considered methods for improving their stability and speed of convergence. Will also address problems with non-linear along the boundaries conditions. As a result, the work will be presented to the most effective method of solving nonlinear problems heat transfer.
Heat and mass transfer processes are dominant in all the ways to obtain energy create a microclimate within the residential and industrial buildings (air conditioning, ventilation, heating - in winter and cooling - in the summer). These processes are developed as indoors and Walling. The walls of residential and public buildings, heat and power units, boilers, methodical and heating furnaces, wells, etc. exposed Environment and lose her heat.
In the design of various thermal power units: boilers, heating and melting furnaces, heat engines, compressors, chillers, process equipment metallurgy, chemical and food industry should take into account the processes of heat transfer, as they become determinant when choosing a design. Feasible and cost-effective will design, which made ??the best thermal regime. Predict such regimes on the natural object is not always convenient and cost- profitable, so the help is very often comes to the similarity theory and subsequent numerical simulation.
Many problems of heat and mass transfer, encountered an engineer not amenable to analytic solution, and the way their theoretical analysis - obtaining numerical solutions.
Analytical methods can solve only a certain class of differential equations partial derivatives with constant transfer coefficients. Obtaining the solution of nonlinear equations in an analytical form for the dependence of transport coefficients of the unknown function is only possible in exceptional cases.
The special difficulty is the so-called external nonlinearity caused by, for example, the presence of radiation energy on the surface of the area. Universal method for finding an approximate solution of differential equations applicable to a wide range of problems in mathematical physics, is the method of finite differences.
The continuous variation of the arguments is replaced by a finite, discrete set of points (nodes), called the grid. Instead of the desired function of continuous arguments are searching function discrete arguments, defined at grid - the grid function. Derivatives in differential equation are replaced (approximated) by the corresponding difference relations, ie linear combination of values ??of the grid function in multiple grid nodes. Thus, the differential equation is replaced by a system of algebraic equations. Boundary (initial and boundary) conditions are also replaced by the corresponding difference conditions for the grid function.
Set of rules of writing difference equations and boundary conditions expressed in the difference form, can be called a difference scheme, and the nodes involved in it - a template. Set of nodes corresponding fixed point in time, called the temporary layer.
The resulting solution of difference problem will approximate the original problem. Obviously, that the transition to a discrete argument implies the desire of the difference problem to the original under refinement of the mesh. In this case, the scheme should guarantee the convergence solution obtained.
Replacing the differential problem in advance of the difference involves the introduction of errors, ie - approximation error. It is characterized by the residual obtained by substituting exact solutions of the original problem in the differential. Such verification is carried out, for example, on the basis of expansion of the exact solution in Taylor series in the grid. Estimate the residual value of the index power grid steps in the discarded terms of the series. Moreover, if the boundary conditions or source term of the original equation is approximated with a different error, that it take into account overall assessment of accuracy.
Until now, used the assumption that the difference in arithmetic problem performed exactly, ie solution can be obtained from any number of significant digits. Almost they are conducted with a finite number of characters. At each stage of computation errors are allowed rounding, which can be regarded as a perturbation of the initial data for subsequent phase calculations.
If small rounding errors in intermediate stages of computations in the grid refinement leads distort the decision to "discrepancies" that such a scheme is called unstable. Therefore, the scheme should guarantee the weak dependence of the resulting solutions of the difference problem on a small change input data. If this requirement is satisfied, then the scheme is called stable.