1549301742-The_Theory_of_Difference_Schemes__Samarskii

34 Prelirninaries

1.2 SOME VARIANTS OF THE ELIMINATION METHOD

1. The flow variant of the elimination method for difference problems with
   widely varying coefficients. In numerical solution of thermal-flow hydrody-
   namics and magnito-hydrodynamics problems in which the coefficients of
   heat conductivity and electric conductivity depend on the thermodynamic
   parameters of the medium the elimination method for the corresponding
   difference equations leads to unsatisfactory results. In thermal problems
   adiabatic cells with infinitely large thermal-flow reveal themseves. The
   nonconductive cells or ideal conductivity may appear in magnetic prob-
   lems, thus causing obstacles in connection with widely varying coefficients
   and considerable accuracy losses. In mastering the difficulties involved,
   the flow variant of the elimination method is aimed at solving a revised
   supplementary problem relating to the heat flow.

( 61)

(62)

The statement of the boundary-value problem is

a; Yi-1 - c; Yi + ai+l Yi+1 = -f; , z = 1, 2, ... , JV - 1,

with the n1embers

c; = a;+1 +a; + d;, d; > 0,^0 <a; < =,

(63)

Having stipulated condition (63), the c01nputational formulae of the right
elimination with regard to problem (61)-(62) can be written as

i = 0, 1, 2, ... , N - 1,

(64) a;+1 +a; ( 1-O:;) +cl;'

CY;+ 1 ·.

~ (a; /3; + f; ) ,

a;+1

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

A new unknown difference function is called a flow and is defined by

(65) w; : = a; ( Yi - 1 - Yi ) ,