54
2. Basic Finite-Difference Methods
other Runge-Kutta scheme is the midpoint method , in which a = ! and ß = 1.
An example of a non-Runge-Kutta scheme is the Matsuno , orfotward-backward.
method, for which a = 1 and ß = 1 (Matsuno 1966b).
If the preceding multistage formu1a is applied to the oscillation equation, the
result is
f/Jn+1 = f/Jn + ßi« !:l.t(f/Jn + ai« !:l.tf/Jn) + Cl - ß)iK !:l.tf/Jn .
The amplification factor is
(2.38)
and
(2.39)
which shows that the set of second-order Runge-Kutta schemes (i.e., those
schemes for which aß = !)have O[(!:l.t)4] amplitude error, whereas the amplitude error in the two-stage non-Runge-Kutta schemes is o[(!:l.t)2]. Unfortunately,
all the Runge-Kutta schemes are unstable, since in the limit of good numerica1
resolution ,
IAIR-K2::::: 1 + k(K!:l.t)4 .
Although the Runge-Kutta schemes are unstab1e, the growth is O[(!:l.t)4] . At a
given step size, the erroneous amplification produced by the second-order RungeKutta methods will be much weaker than the O[(!:l.t)2] growth produced by forward time-differencing (see [2.36]). The slow growth generated by the RungeKutta methods can sometimes be tolerated if K!:l.t is sufficiently small and the
totallength of the integration is sufficiently short.'
Many physical systems contain several different modes, each oscillating at a
different frequency. When simulating these systems , the highest-frequency components of the numerical solution are likely to be most seriously in error because
of their poor numerical resolution. It is precisely these poorly resolved features
that amplify most rapidly in the Runge-Kutta solutions. The amplification of the
highest-frequency components can be prevented by choosing other values for a
and ß, although such a choice also increases the truncation error. According to
(2.39), very low frequency oscillations (K!:l.t « 1) will be stable whenever aß is
greater than !.Matsuno (l966b) suggested setting a = 1, ß = 1, in which case
(2.39) becomes
(2.40)
The Matsuno scheme damps the solution whenever 0 < «S: < 1. Differentiation of (2.40) with respect to K!:l.t shows that the maximum damping occurs when
5As explored in Problem 15, the auxiliary relat ion 0(61) :s O[(ßx)4j3) may be required to
ensure the convergence of finite-difference approximations to the advection equation when the time
difference is evaluated by a second-order Runge-Kutta scheme.
2. Basic Finite-Difference Methods
other Runge-Kutta scheme is the midpoint method , in which a = ! and ß = 1.
An example of a non-Runge-Kutta scheme is the Matsuno , orfotward-backward.
method, for which a = 1 and ß = 1 (Matsuno 1966b).
If the preceding multistage formu1a is applied to the oscillation equation, the
result is
f/Jn+1 = f/Jn + ßi« !:l.t(f/Jn + ai« !:l.tf/Jn) + Cl - ß)iK !:l.tf/Jn .
The amplification factor is
(2.38)
and
(2.39)
which shows that the set of second-order Runge-Kutta schemes (i.e., those
schemes for which aß = !)have O[(!:l.t)4] amplitude error, whereas the amplitude error in the two-stage non-Runge-Kutta schemes is o[(!:l.t)2]. Unfortunately,
all the Runge-Kutta schemes are unstable, since in the limit of good numerica1
resolution ,
IAIR-K2::::: 1 + k(K!:l.t)4 .
Although the Runge-Kutta schemes are unstab1e, the growth is O[(!:l.t)4] . At a
given step size, the erroneous amplification produced by the second-order RungeKutta methods will be much weaker than the O[(!:l.t)2] growth produced by forward time-differencing (see [2.36]). The slow growth generated by the RungeKutta methods can sometimes be tolerated if K!:l.t is sufficiently small and the
totallength of the integration is sufficiently short.'
Many physical systems contain several different modes, each oscillating at a
different frequency. When simulating these systems , the highest-frequency components of the numerical solution are likely to be most seriously in error because
of their poor numerical resolution. It is precisely these poorly resolved features
that amplify most rapidly in the Runge-Kutta solutions. The amplification of the
highest-frequency components can be prevented by choosing other values for a
and ß, although such a choice also increases the truncation error. According to
(2.39), very low frequency oscillations (K!:l.t « 1) will be stable whenever aß is
greater than !.Matsuno (l966b) suggested setting a = 1, ß = 1, in which case
(2.39) becomes
(2.40)
The Matsuno scheme damps the solution whenever 0 < «S: < 1. Differentiation of (2.40) with respect to K!:l.t shows that the maximum damping occurs when
5As explored in Problem 15, the auxiliary relat ion 0(61) :s O[(ßx)4j3) may be required to
ensure the convergence of finite-difference approximations to the advection equation when the time
difference is evaluated by a second-order Runge-Kutta scheme.