A General Scheme of the Method of Difference Potentials for Differential Problems
The scheme for applying the method of difference potentials (MDP) to problems of the form (I), (II), which will be described in this chapter, is a generalization of the scheme that was outlined in the introduction to the book and then realized in detail i
- PDF / 4,972,713 Bytes
- 56 Pages / 439.37 x 666.14 pts Page_size
- 13 Downloads / 171 Views
The scheme for applying the method of difference potentials (MDP) to problems of the form (I), (II), which will be described in this chapter, is a generalization of the scheme that was outlined in the introduction to the book and then realized in detail in Part I for the main boundary-value problems in the case of the Laplace equation on the plane. The generalization consists chiefly in using the construction of the potential described in Part II and based on the notion of a clear trace. We then have to deal with boundary equations with projections in the unknown potential density belonging the space of clear traces. The generalization also consists in the fact that for the discretization of boundary equations with projections we use difference potentials whose const ruction is based on the notion of a clear difference trace. The solution of the resulting overdetermined system of algebraic equations is found by the method of least squares and computed by iterations.
1.1 Nonclassical Auxiliary Problems In constructing an auxiliary problem, we may choose the domain DO and the space UDO more or less arbitrarily, keeping in mind the necessity of obtaining a well-defined auxiliary problem admitting the construction of an approximating stable difference boundary-value problem that can be easily solved. It should be noted that the auxiliary problem in question can be a boundary-value problem having no definite physical meaning by itself, but its importance is due to its use in MDP . The construction of such differential problems and approximating difference problems play the same role in MDP as the construction of analytic expressions for fundamental solutions or Green functions in the classical numerical potential methods. 1.1.1 Examples
Let us illustrate the construction of nonclassical auxiliary problems intended for the solution of int ernal and external boundary-value problems for the Helmholtz equat ion V. S. Ryaben'kii, Method of Difference Potentials and Its Applications © Springer-Verlag Berlin Heidelberg 2002
218
1. A General Scheme of the Method of Difference Potentials
(1.1)
Example 1.1 .1. We shall consider (1.1) in a bounded domain D that lies, together with the boundary r = aD, inside the square DO = {O < Xl> X2 < 11"}. For an auxiliary problem, let us use the classical first boundary-value problem, relating to UDO all sufficiently smooth functions U DO vanishing on the boundary ot». For an approximating difference boundary-value problem, we take the simplest five-point difference analog of this boundary-value problem: Ll(h)U m
+ p,2 Um
,1-
+ Ll x 2 x 2 U m + p,2 Um = 1m, Um 1+l,m2 - 2Umt.m2 + Um 1-l,m2
= Ll x 1x 1U m _
h2
XIXIUm -
,1X2X2
Um
-
Um 1,m2+ 1 -
-
'
2Umt.m2 h2
+ Um 1,m2-1
(1.2)
'
where h = 1I"/N and N is a positive integer. Let us show that this choice of an auxiliary problem can be unfortunate. The operator ,1 + p, 2l considered in the space UD has the eigenfunctions'ljJ(k ,l) and eigenvalues )..k,l which are given by 'ljJ(k,l) )..k,l
= sin kXl sin lX2, = p,2 _ (k 2 + l2
Data Loading...
