1549301742-The_Theory_of_Difference_Schemes__Samarskii

176 Homogeneous Difference Schemes

and having k 1 = 2 and k 2 = 4 to the difference problem with steps h 1 = h

and h 2 =! h we calculate the solutions Yh and Yh/ 2 and the coefficients

c 1 = -!- and c 2 = ~, leading to the function in question:

Y -( x ) =^4 3 Yh/2 - 3 1 Yh ,

which is defined on the grid wh and approximates the exact solution u(x)

with accuracy O(h^4 ):

One way of proceeding is to take for granted the expansion

where the functions cv 1 ( x) and cv 2 ( x) are independent of h. Because of

this, we must solve the difference problem concerned three times with steps

h 1 , h 2 and h 3 , respectively, in an attempt to obtain a solution with the

prescribed accuracy O(hl.:^3 ). In what follows Y1i, (x), yh 2 (x) and yh 3 (x) will

stand for the appropriate solutions. Also, it will be sensible to introduce

their linear cornbination

As before, the grid wh is the intersection of the three grids wh,, wh 2 and

wh 3 • In particular, wh = wh under the agreements h 1 = h, h 2 = !h and

h 3 = ih. When the approximation Yh =uh+ O(hk^3 ) is accepted for later

use, the following equations

constitute the system for finding the coefficients c 1 , c 2 , c 3. Evidently, the

determinant of this system is nonzero.

In the general case the expansion of the error yh - uh in powers of h

is of the form

n-1

(18) Yh = 1lh + L CV 8 (x) h^8 + cvn(x, h) hn,

s=l

where CV 8 ( x ), s = 1, 2, ... , n-l, are independent of h and the absolute value
of cvn(x, h) is bounded by a constant M = const > 0, which is independent
of h. Our next step is to establish an a priori expansion of the form (18).