1549301742-The_Theory_of_Difference_Schemes__Samarskii

Higher-accuracy schemes

We will prove the indicated properties.

1) Property 1) follows directly from problem staternent (5 )-( 6).

2) Taking into account (5) and (6), we have for L = L(p,q)

= -v;(xi+i) + v~(xi_ 1 ).

3. Applying Green's formula on the segment [xi-i, xi] yields

= ~ (v;)'(xi) v~(xi) - ~ (v;)'(x;) v;(x;_ 1 ) + v~(x;_ 1 ).

P; P;

Upon substituting here the relations

Xi

-(v;)'(x;)^1. = 1+ J

Pi

q(x) v; (x) dx,

x;+1

• 1 ( V,: 2 )'( X; ) = -1 -
  Pi

j q(x) v;(x) dx

we establish property 3).

x· l

4) Having stipulated the conditions

the function v;+^1 ( x) does follow

x;+1

0 = j ( v;+^1 L v; - v; L v;+^1 ) dx

xi

1 dvi+1 1 I = 1

p dx x=x,

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