1549301742-The_Theory_of_Difference_Schemes__Samarskii

90 Basic Concepts of the Theory of Difference Schemes

Table 1 contains the values Yi at several nodes of the grid wh for the four

variants. In the final line of this table the numbers of grid nodes connected

with abnormal termination are indicated by "infinitiy".

Table 1

Variant 1 Variant 2 Variant 3 Variant 4

i Yi i Yi i Yi i Yi

6 0.952 7 0.567 32 0.724 28 8.97 .10-^2

7 0.942 8 0.516 33 0.703 29 7.87 -10-^2

11 0.900 10 0.420 37 0.260 32 3.78 .10-^3

12 0.832 (^11) 0.306 38 -0.196 33 -7.41 -10-^2
13 0.174 12 -0.641 39 -1.11 34 -0.237
(^14) -7.05 13 -1.17-10^1 40 -2.95 40 -3.17 · 101
15 -8.72·10^1 14 -1.45. 102 50 -4.10. (^103 50) -7.84·10^4
16 -9.77·10^2 15 -1.76.^103 60 -4.64.^106 60 -1.94 · 108
20 -1.49 · 107 20 -4.54 · 108 80 -5.92 · 1012 80 -1.19-10^15
25 -2.52·10^12 25 -1.17·10^14 90 -6.69 · 1015 90 -2.94·10^18
(^32) = (^30) = (^100) = (^92) =

2. The Cauchy problem for a system of differential equations of first order.
   Stability condition for Euler's scheme. We illustrate those ideas with
   concern of the Cauchy problem for the system of differential equations of
   first order

(6) t > 0, u(O) = u 0 ,

where u = ( u( l), u(^2 ), ... , u(n l) is the vector of unknowns, u 0 = ( uP), u~^2 ),
... , u~n l) is an n-dimensional given vector and A = ( aij) is a symmetric
positive definite n x n-matrix. In what follows a vector space Hn is equipped
with the inner product (u, v) = I:~=l u(ilv(il and associated norm II u II =
J(u, u). Under these structures, a linear self-adjoint operator A: Hn r-+
H n, A = A* > 0, is associated with the matrix A of interest.
We denote by { >.k , ek} the system of eigenvalues and orthonormal
eigenvectors of the operator A:

k = 1,2, ... ,n,