Page 59 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

44 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

a nonlinear equation written in conservative form, Le., in a domain D

of the plane (z,t), namely

ur + (f (u)), =0

Or

divF = 0

where F = (f (uw) ,u) and “div” is the space-time divergence operator.
If ® is a smooth function with compact support in the plane (z,t), then
the above differential equation leads to the fulfilment, for any ® , of the
“orthogonality” relation { @divFdzdt = 0 which comes, by integrating
D
by parts, to [ grad® -Fdzdt =0.
D
If wu is a smooth function the last relation is eguivalent with the given

differential equation; but if it is not smooth enough, the last equality
keeps its sense while the differential equation does not.
We will say that u is a weak solution of the differential equation if it
satisfies { grad® - Fdzdt = 0 , for any smooth function ® with compact
D
support. Obviously, if we want to join also the initial conditions u(z,0) =
uo, then, integrating on D, (a subdomain of D from the half-plane t¢ > 0)
we get

| sraas - Fdzdt + / ® (x, 0) up (x) dz = 0

Dt DNOz

and if ® has its support far from the real axis, the last term would
disappear again.
So we have both a differential and a weak form for the considered
equation. We will also have an integral form if we integrate the initial
b
equation along an interval [a,b] of the real axis, precisely a fudz =
a
f (u)|®.

Of course we should ask if a weak solution satisfies necessarily the in-
tegral form of the equation ? Provided that the same quantities, which
showing up in the conservative form of the equation are kept for the in-
tegral form too, the answer is affirmative. That is why the weak solution
will be basically the target of our searches.
Let us now investigate the properties of the weak solutions of the
conservation law uz + (f (u)),, = 0 in the neighborhood of a jump dis-

continuity (i.e., of first order, the only ones with physical sense). Let u
be a weak solution along the smooth curve © in the plane (z,t). Let ®