1549301742-The_Theory_of_Difference_Schemes__Samarskii

Selected problems

3. The statement of the problem is

OU 82 1t

8t - CJx2, O<x<l, t > 0)

OU

ox (0, t) = μi(t) J

1l ( x, 0) = 1l 0 ( x).

Construct a perfect difference schen1e of accuracy 0( r^2 + h^2 ).

4. Find stability conditions for the difference scheme

z • Yt,;. _ - O" Yx:i:,j n+l + (1 _ ) O" Yxx,j n _ ( q μ Yj n+l + (1 _ ) μ Yn)

1 ,

j=l,2, ... ,N-1, Yo n = YN n = ,^0

approximating the equation

au 82 u

z at = CJx2 - q ·u ) q =canst > 0,

ll(O, t) = u(l, t) = 0,

u (x, 0) = u 0 (x).

5. Determine the exact solutions of the boundary-value problem
   8tt all
   8t +ox=^0 ) O<x<)(, O<t<T,

u (x, 0) = u 0 (x), u (0, t) = v.1(t))

and of the difference schen1e approximating this problem:

-i = 1, 2, ... ) n=0,1,2, ... ,

X; = -ih,

379

Proceeding from the explicit representation of solutions, show that if the
condition I= r/h = canst is fulfilled as T--+ 0, h--+ 0, then convergence
occurs only for r < 1 and the difference problem solution coincides with
the differential problem solution for r = 1.