1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems

with the coefficients

(25')

Xi+l/2

b; = Cl;+ 1 exp { j 1~(t) clt} ,

Xi

x·

a, = Cl; exp {- J r(t) clt}.

Xi-1 /2

187

It is straightforward to verify that the scheme concerned provides an ap-

proximation of order 2 clue to the expansions

bi - ai = k' + r + O(h2)

h z ' '

• b +a
  ''=ki+O(h^2 ).
  2

Replacing the integrals in (25') to 0( h^2 ) by the expressions ~ (3f; + ri+ 1 )

and ~ (3f; +ii_ 1), respectively, we arrive at the monotone scheme with the

coefficients

and

generating an approximation of order 2.

4. Difference schemes for a stationary equation in cylindrical coordinates.
   The stationary diffusion or heat conduction equation

div (k grad u) - qu = -f(r, <p, z)

takes in the cylindrical coordinate system (r, <p, z) the form

(26)

~ :r (~ k(r) ~~) - q(r) u = -f(r), O<r<R,

q(r) > 0,

in the case when the solution u = 1t(r) depends neither on z nor on <p, that
is, in the case of the axial symmetry.
When r = 0 we impose the boundedness condition I u(O) I < oo being
equivalent to the requirements

(27) ,. Jim --+ 0 r k(r) clclu 7' = 0.