1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference sche1nes as operator equations. General formulations 117

1. Difference schemes as operator equations. After replacing differential
   equations by difference equations on a certain grid wh we obtain a system of
   linear algebraic equations that can be written in matrix form. The outcome
   of this is

(1)

where U is a square N x N-matrix, Y = (y 1 , y 2 , .•. , yN) is the vector of

unknowns and <[> = ( <p 1 , <p 2 , ... , 'PN) is a known right-hand side including

the right-hand sides of boundary conditions. With every matrix U one

can associate a linear operator A acting from RN into RN. With this

correspondence in view, equation ( 1) takes the form

(2) Ay=<p,

where the unknown vector y is sought, while the right-hand side 'P E RN is

a given vector. The operator A maps onto itself the space of grid functions

defined on wh and satisfying the h01nogeneous boundary conditions. Several

examples can add interest and aid in understanding.

Example 1. The first boundary-value proble1n. Given on the segment

[O, l] an equidistant grid wh = {x; = ih, i = 0, 1, ... , N, h = 1/ N},

let us look for a solution of the first boundary-value problem

(3) Ay=y_ xx =-f(x), O<x=ih<l,

or

(3')

i = 1, 2, ... , N - 1 ,

As the first step towards the solution of this problem, we form the vector
Y = (y 1 , y 2 , ..• , yN_ 1 ), making it possible to rewrite equation (3) in the
form (1) with the (N - 1) x (N - 1)-matrix

2 -1

-1 2

0 -1

0 0

0 0

-1 0

2 -1

0 0

0

0

0

0

0

0

-1 2