1549301742-The_Theory_of_Difference_Schemes__Samarskii

320 Difference Schemes with Constant Coefficients

Substitution of ( 49) into ( 48) gives

Summing over/= 0, 1, ... ,j and keeping y^0 = 0, we deduce that

In agreement with Lemma 1 in Chapter 2, Section 3, we might have

thereby justifying that

1 [ j ] 1/2

II yJ+i lie < 2 V2f, j't:o^7 II v)' 11

2

,

Applying this estimate to problem (III) we get

thus demonstrating that scheme (III) converges uniformly, so that

(} > ~, u EC;/,

(} = l 2 ' u E C3 '^4

(} = (} * , u E Cg.

A case in point is that for the explicit schen1e with (} = 0 the uniform

convergence does not follow from ( 46) under the constraint T < ~ h^2. But

the a priori estimate emerged in Section 5.7, namely

provides support for the view that

. j I
II z^1 +^1 lie < L^7 111/JJ lie·
j'=O

vVhence it follows imn1ediately that the explicit scheme converges uniformly
with the rate O(r + h^2 ).