1549301742-The_Theory_of_Difference_Schemes__Samarskii

380 Difference Schemes with Constant Coefficients

6. The problem

au

at

u (x, 0) = 0,

u(x,t) = 0, x E 8G, t > 0,

is approximated by the explicit difference scheme on the grid

whT=wh x WT, WT ={tn = nT, r>O},

w h = {x(i) 1 ) x(j)} 2 ) x(i) 1 = ih ll x(j) 2 = J.h 2l

i=l,2, ... ,N 1 -l, j=l,2, ... ,N 2 -l, h 1 N 1 = l 1 , h 2 N2 = l 2.

What inaxirnal step T in tirne should be taken to provide stability of the

scheme when l 1 = 1, l 2 = 10, N 1 = 10, N 2 = 100?

7. Prove for the problem

au

at

au

ox J t > 0,

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

u(O,t)=O,

that the difference scheme

yf+l -y{

T

YI -y{-1
h
is absolutely unstable and the scheme

"j+l Yi - Yi .. J _ "j Yi+l - "j Yi
T h
is stable under the condition T < h.

-00 < x < 0 J

8. Show that for any T and h a pure irnplicit difference scheme (a forward
   difference scheme) approximating the problem

au 82 tt

at - 8x^2 J

tt(x, 0) = tt 0 (x),

O<x<l,

au au

ox (0,t) =ox (l,t) = 0 J

is not asymptotically stable.

t > 0 J