2.3 Time-Differencing
57
The general form for an explicit three-time -level method is
c/Jn+1 = (Xlc/Jn + (X2c/Jn-1 + ßllitF(c/Jn) + ß2/).tF(c/Jn-I).
(2.41)
When formulating a three-level scheme, one seeks to improve upon the two-timelevel methods, so it is reasonable to require that the global truncation error be of
at least second order. The three-level scheme will be of at least second order if
(2.42)
where the coefficient (X2 remains a free parameter. One could choose (X2 to further
reduce the truncation error, but the result is not a useful scheme (see problem 9).
The most important explicit three-level schemes are obtained by choosing (X2 to
minimize the amount of data that must be stored and carried over from the n - I
time level, i.e., by setting (X2 = I, in which case ß2 = 0, or by setting (X2 = O.
If (X2 is set to one, (2.41) becomes the leapfrog scheme. The choice (X2 = 0 gives
the second-order Adams-BashJorth method. The remainder of this section will be
devoted to an examination of these two schemes.
If the leapfrog scheme is applied to the oscillation equation, the result is
(2.43)
Since the preceding is a linear finite-difference equation with constant coefficients, the amplification factor is constant from time step to time step and satisfies
the quadratic equation
A 2 - 2iKlitA - I = O.
The two roots are
A± = i« /).t ± (I - K 2 /).t 2 ) 1/2 .
In the limit of good numerical resolution, K/).t
0 and A+
I; A_
(2.44)
- 1. Ev -
idently, the numerical solution is capable of behaving in two very different ways,
or modes. The mode associated with A+ is known as the physical mode because
it approximates the solution to the original differential equation. The mode associated with A_ is referred to as the computational mode, since it arises solely as
an artifact of the numerical computation . If IK lit 1 :::: I, the second term in (2.44)
is real and IA+I = lA _I = I ; i.e., both the physical and the computational modes
are stable and neutral. In the case K /).t > I,
IA+I = liK/).t + i (K 2lit2 - I) 1
1 2 > liKlitl > I,
and the scheme is unstable. When K /).t < -I, a similar argument shows that
lA_I> 1. Note that when K/).t > I, A+ lies on the positive imaginary axis in the
complex plane, and thus each integration step produces a 90° shift in the phase of
the oscillation. As a consequence, unstable leapfrog solutions grow with aperiod
of 4lit.
The complete leapfrog solution can typically be written as a linear combination
of the physical and computational modes. An exception occurs if Klit = ± I , in
57
The general form for an explicit three-time -level method is
c/Jn+1 = (Xlc/Jn + (X2c/Jn-1 + ßllitF(c/Jn) + ß2/).tF(c/Jn-I).
(2.41)
When formulating a three-level scheme, one seeks to improve upon the two-timelevel methods, so it is reasonable to require that the global truncation error be of
at least second order. The three-level scheme will be of at least second order if
(2.42)
where the coefficient (X2 remains a free parameter. One could choose (X2 to further
reduce the truncation error, but the result is not a useful scheme (see problem 9).
The most important explicit three-level schemes are obtained by choosing (X2 to
minimize the amount of data that must be stored and carried over from the n - I
time level, i.e., by setting (X2 = I, in which case ß2 = 0, or by setting (X2 = O.
If (X2 is set to one, (2.41) becomes the leapfrog scheme. The choice (X2 = 0 gives
the second-order Adams-BashJorth method. The remainder of this section will be
devoted to an examination of these two schemes.
If the leapfrog scheme is applied to the oscillation equation, the result is
(2.43)
Since the preceding is a linear finite-difference equation with constant coefficients, the amplification factor is constant from time step to time step and satisfies
the quadratic equation
A 2 - 2iKlitA - I = O.
The two roots are
A± = i« /).t ± (I - K 2 /).t 2 ) 1/2 .
In the limit of good numerical resolution, K/).t
0 and A+
I; A_
(2.44)
- 1. Ev -
idently, the numerical solution is capable of behaving in two very different ways,
or modes. The mode associated with A+ is known as the physical mode because
it approximates the solution to the original differential equation. The mode associated with A_ is referred to as the computational mode, since it arises solely as
an artifact of the numerical computation . If IK lit 1 :::: I, the second term in (2.44)
is real and IA+I = lA _I = I ; i.e., both the physical and the computational modes
are stable and neutral. In the case K /).t > I,
IA+I = liK/).t + i (K 2lit2 - I) 1
1 2 > liKlitl > I,
and the scheme is unstable. When K /).t < -I, a similar argument shows that
lA_I> 1. Note that when K/).t > I, A+ lies on the positive imaginary axis in the
complex plane, and thus each integration step produces a 90° shift in the phase of
the oscillation. As a consequence, unstable leapfrog solutions grow with aperiod
of 4lit.
The complete leapfrog solution can typically be written as a linear combination
of the physical and computational modes. An exception occurs if Klit = ± I , in