The Method of Finite Differences
This method of investigating various problems for differential equations consists of reducing them to systems of algebraic equations in which the unknowns are the values of grid functions u Δ at the vertices of the grids ΩΔ, and then examining the limit process when the lengths of the sides of the cells in the grid tend to zero. This brings us to the aim if, in the limit, the functions u Δ, give us the solution of the original problem. Such a reduction of the problem to an infinite sequence of auxiliary finite-dimensional problems defining approximate solutions u Δ is neither unique nor uniform for problems of various types. In other words, for every problem we can construct different difference schemes converging to it, and for problems of various types such schemes differ essentially from each other. In this chapter we shall consider the same boundary value and initial-boundary value problems for the equations of the basic types that we studied in the earlier chapters, and for each of them construct a few elementary difference schemes, which give us in the limit the solutions of these problems. This will be done in such a way that the existence of these solutions will not be stipulated in advance, but, to the contrary, will be established by the method of finite differences. We shall restrict ourselves to “rectangular ” lattices in which the cells will be parallelopipeds with faces parallel to the coordinate planes.
KeywordsFinite Difference Difference Scheme Strong Convergence Grid Function Uniform Boundedness
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