1549301742-The_Theory_of_Difference_Schemes__Samarskii

196 Homogeneous Difference Sche1nes

It follows from the foregoing that

,/ * y). -~ -, μi

1 r. z

An estimate similar to that established for problem ( 41) is valid for a solu-

tion of problem (55) with the right-hand side

yielding

This means that scheme (51), (53), (54) is of second-order accuracy if we

agree to consider k(x), q(x), f(x) E C(^2 l[O, 1].

5. Difference schemes for an equation in spherical coordinates. If a solution
   to the equation
   div (kgrad u) - qu = -f(1·, G, tp)

in the spherical coordinate system is centrally symmetric, that is, is inde-

pendent of G and tp, then the function 1l = 11(r) satisfies the equation

(56)

• -^1 d ( r k(r)^2 - du) - q(r) u = -f(r),
  r^2 dr dr
  O<r<R,

q(r)>O.

In the general setting the function u(r) is supposed to be bounded at the

point r = 0. This property is equivalent to the condition

(57) r^2 k(r)-du [ =0.

dr r=O

At the point r = R we may i1npose, for instance, the standard condition

(58) 1l(R) = P2 ·

A bounded solution of problem (56)-(58) possesses the same prop-
erties as in the case of the axial symn1etry (for more detail see problem
(26)-(28)).