1549301742-The_Theory_of_Difference_Schemes__Samarskii

632 Econmnical Difference Sche1nes for Multidhnensional Proble1ns

"Locally one-dimensional" schemes for hyperbolic equations acquires

the form

(69) Ytafa = crp Acr (Y(cr) + Y(cr)) + 2 crp 'Per' CY= 1, ... ,p' p = 2, 3'

where

1/4 for p = 2,

1/3 for p = 3,

and the left-hand side Ytafa is given by forn1ula (67) for p = 2 and by

formula (68) for p = 3. With the detailed fonns in 1nind, the final schen1es

refer to three-point additive ones (p = 2) ancl four-point ones (p = 3).

The principal difference from the case of parabolic equations lies in the

dependence on the number of measurements p.

Equation (69) can be rewritten as

The solution Y(cr) is sought fro1n the three-point equation

for

for

P-2 - '

p = 3.

along seg1nents parallel to the axis Oxcr with the boundary condition

(71) for x E /~

with further reference to the eli1nination method.

First of the initial conditions u(x, 0) = u 0 (:r) is approximated exactly:

(72) y( x, 0) = tt 0 ( X).

The intermediate values y^112 = y(x, r/2) for p = 2 and y^113 = y(x, r/3),
y^213 = y( ;r;, 2r /3) for p = 3 are found, respectively, frmn the equations

(73)

2

( E- T^4 Ai ) y 1/2 = F1,

T_ T

2

F1 = u 0 + - tt 0 + - A1 U 0 + T 2(· f 1 - 1( - A 1l + f ))

2 4 8 t=O