5.1 Conservation Lawsand Weak Solutions
243
associated with advective-form finite differencing are even more apparent if (5.1)
is approximated using the scheme
A,n+ I
'l'j
A,n
-'I'j
!!.1
»» A,n
(n)2'1'j-'I'j-1 =0
+ f/Jj
!!.x
and the initial data
if j s jo,
otherwise.
In this case , the finite-difference approximation to 1/1 281/1 18x is zero at every
grid point, and the numerical solution is stationary. In order to understand how
advective-fonn finite-difference approximations can converge to invalid solutions
to the generalized Burgers's equation, it is helpful to review the sense in which
discontinuous functions constitute solutions to partial differential equations.
5.1 Conservation Laws and Weak Solutions
Many of the partial differential equations arising in fluid dynamics can be expressed as a system of conservation laws of the form
-
8u
+ L.." -fj(u) = 0,
, , 8
81
. 8xj
J
which states that the rate of change of U at each point is detennined by the convergence of the fluxes f j at that point. An example of this type is provided by
the one-dimensional shallow-water equations. Let u denote the velocity and h the
fluid depth, and suppose that there is no bottom topography; then conservation of
mass requires
8h
-
8
+ -(hu) =0,
81
8x
and conservation of momentum implies
8
8 (
h
2
- (hu) + - hu
2
+ g - ) = O.
81
8x
2
If a function contains a discontinuity, it cannot be the solution to a partial differential equation in the conventional sense, because derivatives are not defined
at the discontinuity. Instead, the solution is required to satisfy a family of related
integral equations. Consider solutions to the scalar conservation law
81/1
8
-+-/(1/1)=0
(5.6)
81
8x