35
where G is the elementary solution of the previous initial-boundary value problem. More
precisely G(x,., t,.) satisfies
oG
or (x, y, t, r) + u(y, r).V' I/G(x, y, t, r) - f3IlI/G(x, y, t, r) +
aG(x,y,t,r) = c5(t-r)c5(x-y) inQ vr ,
oG
On (x,y,t,r) = 0 on E vr ,
1/
G(:t,y,t,O)=O on 01/.
Notice that G is regular for y =J :t and r =J t, therefore
lim ZnhIAx[oTj = fT m(r)V'lIG(.,b,.,r).hdr =
n-+oo
I
10
fT
-
a
. -
V'lI(Jo m(r)G(.,b, .,r).hdr = 8bCf.(x,t).h In C(A x [O,T].
(3.109)
(3.110)
(3.111)
(3.112)
On the other hand, by taking 'IjJ(x,t) = -m(t)V'6,,(x - b).h in (3.77)-(3.79) it yields
tP = Znh and then, for z = Znh, (3.102) gives
- fT f _ p(:t, t)V'6,,(x - b).hd:tdt = (fI, ZnhIAx[OTj).
Jo JB(b.~)
•
(3.113)
Finally, as p is regular in B(b,~) for n large enough, we also have
- fT { _ p(x, t)V'6,,(x - b).hdxdt = fT (
V' ",p(x, t).h6,,(x - b)dxdt (3.114)
Jo JB(b.~)
Jo JB(r,.~)
The result follows by passing to the limit as n ---> oo.D
A similar problem for the steady state equation has been considered in Bermudez,
Martinez and Rodriguez[1991] where numerical results are included.
Acknowledgements.- This paper contains research which has been done in collaboration with C. Rodriguez, M.E. Vazquez-Cend6n, M.E. Vazquez-Mendez and M.A. Vilar
at the University of Santiago, Spain.
REFERENCES
ABBOT [1985]: Computational Hydraulics,Pitman, Boston.
ALCRUDO, F., GARciA-NAVARRO, P. AND SAVIRON, J.M. [1993]: Flux-difIerencesplitting for 1D open channel flow equations, Int. J. Num. Methods in Eng. 14, 1009-1018.
AMES, W.F. [1988]: Analysis of mathematical models for pollutant transport and dissipation, Comput. Math. with Appl. 16, 939-985.
where G is the elementary solution of the previous initial-boundary value problem. More
precisely G(x,., t,.) satisfies
oG
or (x, y, t, r) + u(y, r).V' I/G(x, y, t, r) - f3IlI/G(x, y, t, r) +
aG(x,y,t,r) = c5(t-r)c5(x-y) inQ vr ,
oG
On (x,y,t,r) = 0 on E vr ,
1/
G(:t,y,t,O)=O on 01/.
Notice that G is regular for y =J :t and r =J t, therefore
lim ZnhIAx[oTj = fT m(r)V'lIG(.,b,.,r).hdr =
n-+oo
I
10
fT
-
a
. -
V'lI(Jo m(r)G(.,b, .,r).hdr = 8bCf.(x,t).h In C(A x [O,T].
(3.109)
(3.110)
(3.111)
(3.112)
On the other hand, by taking 'IjJ(x,t) = -m(t)V'6,,(x - b).h in (3.77)-(3.79) it yields
tP = Znh and then, for z = Znh, (3.102) gives
- fT f _ p(:t, t)V'6,,(x - b).hd:tdt = (fI, ZnhIAx[OTj).
Jo JB(b.~)
•
(3.113)
Finally, as p is regular in B(b,~) for n large enough, we also have
- fT { _ p(x, t)V'6,,(x - b).hdxdt = fT (
V' ",p(x, t).h6,,(x - b)dxdt (3.114)
Jo JB(b.~)
Jo JB(r,.~)
The result follows by passing to the limit as n ---> oo.D
A similar problem for the steady state equation has been considered in Bermudez,
Martinez and Rodriguez[1991] where numerical results are included.
Acknowledgements.- This paper contains research which has been done in collaboration with C. Rodriguez, M.E. Vazquez-Cend6n, M.E. Vazquez-Mendez and M.A. Vilar
at the University of Santiago, Spain.
REFERENCES
ABBOT [1985]: Computational Hydraulics,Pitman, Boston.
ALCRUDO, F., GARciA-NAVARRO, P. AND SAVIRON, J.M. [1993]: Flux-difIerencesplitting for 1D open channel flow equations, Int. J. Num. Methods in Eng. 14, 1009-1018.
AMES, W.F. [1988]: Analysis of mathematical models for pollutant transport and dissipation, Comput. Math. with Appl. 16, 939-985.