530 Difference Methods for Solving Nonlinear Equationswhere integration is accomplished along a closed curve in the plane (s, t).
To make our exposition more transparent, we introduce the gridswh = {si = ih, i = 0, 1, ... , N, hN = M},
retaining the notations u, r1, p, E with respect to difference equations and
regarding the function v to integer points s = s; on the grid w;, and p, 11, E- to half-integer points s = si+i/ 2 on the same grid.
Other ideas are connected with setting equation (22) in the rectangle
{si-1/2 < S < Si+l/2' tj < t < tj+l}:
(^8) i+ 1/2 tj +l
.I (u^1 +^1 -v^1 )ds+ J (P1+1;2-Pi-l/2)dt=O
(^8) i-l/2 tj
and writing equations (23)-(24) in another rectangle {si < s < si+l' t 1 <
t < i1+1}:
s· z t J.
(^8) i+l tj+l
.I [(E+0.5v^2 )i+^1 -(::+0.5v^2 )i]ds+ j [((pv)i+ 1 -(pv);)]dt=0.
t J.
The integrals built into these identities are replaced by the newly fonned
expressions
ij + l
J p dt:::::: p(oi)T,
tj
where
t j +l
j vdt:::::: vC^02 lT,
tj
tj+l
j (pv)idt=p~~^3 )vi"')T,
tj
a:=l,2,3,4,