160
3. Beyondthe One-WayWaveEquation
Here Xo is the x -intercept of the curve at t = O. Since 1/1 = J, 1/1 must also be
constant along each characteristic curve, and every characteristic is therefore a
straight line , In any region where a1/llax is negative, the characteristics will converge , and for some sufficiently large value of t , these converging characteristics
must cross. At those points where two (or more) characteristics intersect, the solution is multivalued and exhibits a discontinuity, or shock. If the initial condition
is smooth, the time when the solution first develops a shock lc can be determined
by examining the rate at which gradients of 1/1 steepen. Define Sex , t) = a1/l/ax
and note that
dS = as + 1/1 as = (a1/l + 1/1 a1/l) _ (a1/l)2 = _S2.
dt
tu
ax ax at
ax
ax
Integration of the preceding yields
(3.114)
The first discontinuity, or shock, develops when S becomes infinite at a time
t c = -S(xo, 0)-1 determined by the most negative initial value of a1/l/ax. This
behavior may be compared with that for the linear problem with variable coefficients shown in Fig. 3.lOab, in which the characteristic curves never cross (but
rather approach the lines x = kand x = asymptotically) and true discontinuities do not develop over any finite time interval.
Suppose that solutions to Burgers's equation are sought on the periodic domain
o :5 x :5 1 subject to the initial condition 1/I(x, 0) = sin(21l'x) . When (3.113) is
approximated by the advective-form differential-difference equation
difJi +ifJ ' (ifJi+1 -ifJi-l) = 0
(3.115)
dt
J
'
with
= 1/50, the numerical solution appears as shown in Fig. 3.12. 7 The
numerical solution provides a good approximation to the true solution at t = 0.13 ,
at which time the true solution is still smooth and easy to resolve on the discrete
grid. But by t = 0.22 the true solution has developed a shock, and the numerical
solution misrepresents the shock as a steep gradient bounded by aseries of 1argeamplitude short-wave1ength perturbations. These short-wavelength perturbations
are amplifying rapidly and, as a consequence, lIifJII2 is growing without bound.
The growth in lIifJ 112 is an instability, since the t2-norm of the true solution does
not increase with time . If the solution is smooth, 111/1 112 is conserved along with all
other moments, i.e.,
1
1
[1/1 (x)]P dx = 0
(3.116)
7The solution shown in Fig. 3.12 was obtained using a fourth -order Runge-Kutta method and a
very small time step to accurately approximate the time derivative in (3.1 15).
3. Beyondthe One-WayWaveEquation
Here Xo is the x -intercept of the curve at t = O. Since 1/1 = J, 1/1 must also be
constant along each characteristic curve, and every characteristic is therefore a
straight line , In any region where a1/llax is negative, the characteristics will converge , and for some sufficiently large value of t , these converging characteristics
must cross. At those points where two (or more) characteristics intersect, the solution is multivalued and exhibits a discontinuity, or shock. If the initial condition
is smooth, the time when the solution first develops a shock lc can be determined
by examining the rate at which gradients of 1/1 steepen. Define Sex , t) = a1/l/ax
and note that
dS = as + 1/1 as = (a1/l + 1/1 a1/l) _ (a1/l)2 = _S2.
dt
tu
ax ax at
ax
ax
Integration of the preceding yields
(3.114)
The first discontinuity, or shock, develops when S becomes infinite at a time
t c = -S(xo, 0)-1 determined by the most negative initial value of a1/l/ax. This
behavior may be compared with that for the linear problem with variable coefficients shown in Fig. 3.lOab, in which the characteristic curves never cross (but
rather approach the lines x = kand x = asymptotically) and true discontinuities do not develop over any finite time interval.
Suppose that solutions to Burgers's equation are sought on the periodic domain
o :5 x :5 1 subject to the initial condition 1/I(x, 0) = sin(21l'x) . When (3.113) is
approximated by the advective-form differential-difference equation
difJi +ifJ ' (ifJi+1 -ifJi-l) = 0
(3.115)
dt
J
'
with
= 1/50, the numerical solution appears as shown in Fig. 3.12. 7 The
numerical solution provides a good approximation to the true solution at t = 0.13 ,
at which time the true solution is still smooth and easy to resolve on the discrete
grid. But by t = 0.22 the true solution has developed a shock, and the numerical
solution misrepresents the shock as a steep gradient bounded by aseries of 1argeamplitude short-wave1ength perturbations. These short-wavelength perturbations
are amplifying rapidly and, as a consequence, lIifJII2 is growing without bound.
The growth in lIifJ 112 is an instability, since the t2-norm of the true solution does
not increase with time . If the solution is smooth, 111/1 112 is conserved along with all
other moments, i.e.,
1
1
[1/1 (x)]P dx = 0
(3.116)
7The solution shown in Fig. 3.12 was obtained using a fourth -order Runge-Kutta method and a
very small time step to accurately approximate the time derivative in (3.1 15).