Exact method to solve finite difference equations of linear heat transfer problems
Аннотация
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 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 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. 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 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.