534 Difference Methods for Solving Nonlinear Equationswithin which the third equation can be replaced by one of the following
equations:(37)(38)Observe tha(. instead of the latter equation there seem to be at least two
a.!ternatives(39) (E + O ·^5 v(+l)t^2 ) = -(pC^01 l v<o 5)) s'
( 40)Equation ( 39) can be derived from the foregoing by involving equation (32)
at the (i + 1 )th node(41) (^0) .5 ( vi+l^2 ) t -- -vi+l (0.5) Ps,i (01) or (^0) · 5( v(+l)^2 ) -t - - v(+l) (0.5) Ps (01) ·
Combination of (38) and (39) gives immediately ( 40).
Further comparison of ( 40) with (27) shows that a new family of fully
conservative schemes is contained in fa1nily (25 )-(27) of describing conser-
vative schemes with four paran1etel'S as a result of en1ploying the integro-
interpolation inethod.
Another conclusion can be drawn from the preceding that for any er 1
scheme (36) generates an approximation of O(r + h^2 ) and for er 1 = 0.5 it
provides an approximation of O(r^2 + h^2 ), that is, only one scheme of the
fonn (36) can guarantee a second-order approximation in r:
( 42)
With the psevdoviscosity in view, we replace everywhere in formulas
(36) the pressure p by the approved rule g = p + w, leaving us with
( 43) ·u t = -g(o1) g ' 7] t = v(0.5) s ' E t = _, 9 (o^1 )v(0.5) s , g=p + w.
For the ideal gas and the linear viscosity, this amounts to
p17=(1-l)E,
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( 44) w = -- 17 v s '