1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some variants of the elimination method 37

•successive refinement of the coefficients for z = 1, 2, ... , N - 1 with the

aid of the relations

{

(Cl'; + cl;) ( 1 + (Cl' i + cl;) ai+\ )-^1 , a;+1 > 1,

Cl'i+ l

ai+l (Cl'i + cli) ( ai+l +(Cl;+ cli))-

1

, ai+l < 1 J

Ii+! = {

(Ii+ Ji) ( 1 + (Cl1 + cli) ai+\ )-

1

, a;+1 > 1 J

ai+l (Ii+ f;) ( ai+l + (Cl'i + cli))-

1

, ai+l <^1 · '

• revision of proper boundary conditions

IN (1-X2)-(]'N V2

(^1) - X2 + X2 (]'NaN -1 '
IN ( 1 - X2) Cl N - Cl' N v 2 Ct N
a N ( 1 - X2) + X2 Cl' N

• successive determination of solutions for i = N - 1, N - 2, ... , 0:

Wi

(1-

--Cl'i+l) Yi+! + --Ii+! , ai+1 > 1 ,

a;+1 ai+l

a;+1 Ii+ fi

-------Yi+! + , a1+1 < 1.

a1+ 1 + Cl'i + cl1 a;+ 1 + Cl'; + cl;

Yi

Let us stress that the computational algorith111 given above is stable.

2. "Cyclic elimination method. We now focus the reader's attention on
   periodic solutions to difference schemes or systems of difference schemes
   being used in approximating partial and ordinary differential equations in
   spherical or cylindrical coordinates. A system of equations such as

(76) ai Yi-1 - Ci Yi+ b; Yi+1 = -f;, i = 2, 3, ... , N - 1,