540 Difference Methods for Solving Nonlinear Equations
The preceding is equivalent to the inequalities
(58)
thus causing, in fact, some restrictions on the step in time in connection
with the dependence upon the variations of volume 17 (or density p = 1/17).
By setting k = 0 and choosing ~ = 77 we obtain
or, what an1ounts to the same,
(59) (1 - q (1 - b)) 1) < ~ < (1 + q (1 - b)) 17'
When the isotermic flow of the ideal gas ( 17) is considered, scheme ( 43)
can be written in simplified form, since the energy equation was disappeared
because T = const. A proper iteration process is governed by the same
rule as in the adiabatic case, the convergence of which can be established in
a similar way without difficulties. In the isotermic case with the assigned
values I= 1, a= <XJ, b = 0 we deduce instead of (58) that
1) k i/
--<77<--
l+q - 1-q
and
17
Under the first or second condition (57) the following relations occur:
k+I k IJ
llb g lie < q +i llb g lie,
thereby justifying the convergence rate as a geometric progression for the
iterations just established.
Numerical calculations for I = 5 /3 (a = 1.5) showed that the itera-
tions within the framework of Newton's method converge even if the steps
T are so large that the shock wave runs over two-three intervals of the
grid wh in one step T. Of course, such a large step is impossible from a
computational point of view in connection with accuracy losses. Thus, the
restrictions imposed on the step T are stipulated by the desired accuracy
rather than by convergence of iterations.