142
3. Beyond the One-Way WaveEquation
tion to (3.75) whenever the true solution is bounded. An ideal scheme for the
solution of (3.75)would be A-stable , accurate, and efficient. Both the backward
and trapezoidal time-difference approximations to (3.75) are A-stable. The backward and trapezo idal methods are also implicit; there are no A-stable explicit
schemes. Although the trapezoidal method is stable and accurate, it can be inefficient to approximate every term in the goveming equations using trapezoidal
time-differencing. Some of the terms that generate oscillatory forcing in practical applications are nonlinear (e.g., the nonlinear advection operator v . "i/v in the
momentum equations (1.31», and in order to avoid solving nonlinear implicit algebraic equations on every time step, the nonlinear advection terms are usually
approximated using an explicit finite difference. Consider, therefore, a pair of hybrid schemes in which the oscillatory forcing in (3.75) is integrated using leapfrog
differencing, but the dissipation term is approximated using either back ward or
trapezoidal differencing.
If (3.75) is approximated using the leapfrog-backward scheme
A,n+1
'I'
A,n-I
- ' I '
(3.78)
=iwrjJn+J...rjJn+l,
2111
the combinations of ). and iiJ that yield stable physically reasonable solutions satisfy
iiJ2 < 1 - 2). and ). O.
This region lies within and to the left of the curves labeled LFB in Fig. 3.7a. The
region of useful stability for the 1eapfrog-trapezoidal method,
A,n+1 A,n-I
J... (
)
- 'I'
= iwrjJn + _ rjJn+1 + rjJn-1
(3.79)
'I'
2111
2
'
is
iiJ2 + P < 1 and ). s 0,
and lies within the curve labeled LFT in Fig. 3.7a. Since damping reduces the
amplitude ofthe true solution to (3.75), and since the damping term in both (3.78)
and (3.79) is approximated by an A-stable scheme, one might hope that the stabil -
ity condition for the leapfrog-differenced purely oscillatory problem would be a
sufficient condition for the stability of the full hybrid scheme. This is the case for
the leapfrog-backward method, which is stable whenever Iwl11I < 1. On the other
hand, in order to maintain the usefu1 stability of the leapfrog-trapezoidal method,
the time step must be reduced as the damping rate IJ...I increases, and whenever
J...111 < -1 the method is either unstable or exhibits 4111 oscillations independent
of the actual value of w111 .
The leapfrog-backward scheme is not a particular1y attractive method because
it is only first-order accurate, yet being implicit, it requires essentially the same
computational overhead as the more accurate trapezoidal method. The stability
and accuracy potentially available through a trapezoidal approximation to the
damping term can be better realized if the leapfrog approximation to the oscilla-
3. Beyond the One-Way WaveEquation
tion to (3.75) whenever the true solution is bounded. An ideal scheme for the
solution of (3.75)would be A-stable , accurate, and efficient. Both the backward
and trapezoidal time-difference approximations to (3.75) are A-stable. The backward and trapezo idal methods are also implicit; there are no A-stable explicit
schemes. Although the trapezoidal method is stable and accurate, it can be inefficient to approximate every term in the goveming equations using trapezoidal
time-differencing. Some of the terms that generate oscillatory forcing in practical applications are nonlinear (e.g., the nonlinear advection operator v . "i/v in the
momentum equations (1.31», and in order to avoid solving nonlinear implicit algebraic equations on every time step, the nonlinear advection terms are usually
approximated using an explicit finite difference. Consider, therefore, a pair of hybrid schemes in which the oscillatory forcing in (3.75) is integrated using leapfrog
differencing, but the dissipation term is approximated using either back ward or
trapezoidal differencing.
If (3.75) is approximated using the leapfrog-backward scheme
A,n+1
'I'
A,n-I
- ' I '
(3.78)
=iwrjJn+J...rjJn+l,
2111
the combinations of ). and iiJ that yield stable physically reasonable solutions satisfy
iiJ2 < 1 - 2). and ). O.
This region lies within and to the left of the curves labeled LFB in Fig. 3.7a. The
region of useful stability for the 1eapfrog-trapezoidal method,
A,n+1 A,n-I
J... (
)
- 'I'
= iwrjJn + _ rjJn+1 + rjJn-1
(3.79)
'I'
2111
2
'
is
iiJ2 + P < 1 and ). s 0,
and lies within the curve labeled LFT in Fig. 3.7a. Since damping reduces the
amplitude ofthe true solution to (3.75), and since the damping term in both (3.78)
and (3.79) is approximated by an A-stable scheme, one might hope that the stabil -
ity condition for the leapfrog-differenced purely oscillatory problem would be a
sufficient condition for the stability of the full hybrid scheme. This is the case for
the leapfrog-backward method, which is stable whenever Iwl11I < 1. On the other
hand, in order to maintain the usefu1 stability of the leapfrog-trapezoidal method,
the time step must be reduced as the damping rate IJ...I increases, and whenever
J...111 < -1 the method is either unstable or exhibits 4111 oscillations independent
of the actual value of w111 .
The leapfrog-backward scheme is not a particular1y attractive method because
it is only first-order accurate, yet being implicit, it requires essentially the same
computational overhead as the more accurate trapezoidal method. The stability
and accuracy potentially available through a trapezoidal approximation to the
damping term can be better realized if the leapfrog approximation to the oscilla-