1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schemes of nonstationary gas dynamics 537

k k k k k

f 4 = E - a g 7] - a IJ v 8 ,

Ll A kl]+l = k+l r/ - 7] k , etc.

D urmg. the course o f the ellmmat1on... o f Ll A l.;+l E , Ll A k+l 77 anc j Ll A k+l v f ron1

the foregoing we obtain the three-point difference equation for cletermina-

k+1 k+l

tion of y = 6. !J

k k /,; k k

where F is expressed in terms of f 1 , f 2 , f 3 and f 4. The elimination

method can be en1ployed for the last equation, permitting one to find,

first, 6. 1.:+1 v , 6. k+l 7] , 6. k+l E with. knowledge of 6. k+l g k+l y and, second,

k+l g -_ Ll A k+l g + g k , k+l ·u -_ Ll A k+l v + v k , etc.

6. Convergence of Newton's method. We are now in a position to find out
   the conditions under which Newton's method converges. With this a1m,
   the differences

/). k+l k+l /). k+l k+l '''v '+1 -- ~v '·+1 - v ,

g = g - g , 1/ = 17 - 7] , u

where g, i) and u are the exact. solutions to equations ( 4:3), will be given
special investigation. For this, we write down a typical equation related
to such a difference. By the linearity of equations ( 46) the homogeneous
equations

( 49) 0 k+l 1/ =0.5TOvk+l 8 , k=0,1,2,. .. ,

follow im1nediately fron1 the fort>going. Putting equations ( 4 7)-( 48) to-
gether with the newly formed differences

Ll A A"+l v = u ;: k+l ·u - u ;: v k ~g k+l =b k+l g -bg, k