3.4 Diffusion, Sourees, and Sinks
137
implying that the correct amplification factor for the kth mode is
bk(t +
bk(t)
= e-Mk2M,
which (for M > 0) is areal number between 0 and 1. However, if i < v S !,the
numerical amplification factor for the
as a consequence, the sign of the
mode lies in the interval [-1, 0), and
mode flips every time step as it gradually
damps toward zero. If one wishes to avoid "over damping" the poorly resolved
modes ,
must satisfy the more restrictive criterion that 0 S v si.
As is the case with numerical approximations to the advection equation, the
stability of an explicit finite-difference approximation to the diffusion equation is
limited by the
mode. The behavior of the
mode relative to the longer
modes in the advection problem is, however, quite different from that in the diffusion problem. In the advection problem the shortest waves translate without loss
of amplitude (and may even amplify as the result of deformation in the wind field
or nonlinear processes), so any errors in the simulation of the short waves can
have a serious impact on the accuracy of the overall solution. On the other hand,
as implied by (3.70), diffusion preferentially damps the shortest modes, and after
a brief time the amplitude in these modes becomes negligible relative to that of
the total solution. Since the accuracy with which the short waves are simulated is
irrelevant once those waves have dissipated, an acceptable approximation to the
overall solution can often be obtained without accurately simulating the transient
decay of the most poorly resolved initial perturbations. It can therefore be very
advantageous to approximate the diffusion equation using unconditionally stable
schemes like the trapezoidal method, for which the time step is limited only by
accuracy considerations. In particular, the time step can be chosen to accurately
simulate the transient decay of the physical scales of primary interest, while any
inaccuracies generated by this time step in the poorly resolved modes are hidden
by their rapid decay.
If the time-differencing in the finite-difference approximation to the diffusion
equation is trapezoidal, the resulting scheme
(3.71)
is known as the Crank-Nicolson method. The amplification factor for this scheme
is
1 - v(1 -
Ak = 1 + v(1 -
,
and since M > 0, it follows that JAkl S 1 and the scheme is stable for all M.
Assuming that boundary conditions are specified at the edges of the spatial domain, (3.71) constitutes a tridiagonallinear system for the unknown eP'J+I, which
can be solved with minimal computational effort as discussed in the Appendix.
Although the Crank-Nicolson approximation to the one -dirnensional diffusion
137
implying that the correct amplification factor for the kth mode is
bk(t +
bk(t)
= e-Mk2M,
which (for M > 0) is areal number between 0 and 1. However, if i < v S !,the
numerical amplification factor for the
as a consequence, the sign of the
mode lies in the interval [-1, 0), and
mode flips every time step as it gradually
damps toward zero. If one wishes to avoid "over damping" the poorly resolved
modes ,
must satisfy the more restrictive criterion that 0 S v si.
As is the case with numerical approximations to the advection equation, the
stability of an explicit finite-difference approximation to the diffusion equation is
limited by the
mode. The behavior of the
mode relative to the longer
modes in the advection problem is, however, quite different from that in the diffusion problem. In the advection problem the shortest waves translate without loss
of amplitude (and may even amplify as the result of deformation in the wind field
or nonlinear processes), so any errors in the simulation of the short waves can
have a serious impact on the accuracy of the overall solution. On the other hand,
as implied by (3.70), diffusion preferentially damps the shortest modes, and after
a brief time the amplitude in these modes becomes negligible relative to that of
the total solution. Since the accuracy with which the short waves are simulated is
irrelevant once those waves have dissipated, an acceptable approximation to the
overall solution can often be obtained without accurately simulating the transient
decay of the most poorly resolved initial perturbations. It can therefore be very
advantageous to approximate the diffusion equation using unconditionally stable
schemes like the trapezoidal method, for which the time step is limited only by
accuracy considerations. In particular, the time step can be chosen to accurately
simulate the transient decay of the physical scales of primary interest, while any
inaccuracies generated by this time step in the poorly resolved modes are hidden
by their rapid decay.
If the time-differencing in the finite-difference approximation to the diffusion
equation is trapezoidal, the resulting scheme
(3.71)
is known as the Crank-Nicolson method. The amplification factor for this scheme
is
1 - v(1 -
Ak = 1 + v(1 -
,
and since M > 0, it follows that JAkl S 1 and the scheme is stable for all M.
Assuming that boundary conditions are specified at the edges of the spatial domain, (3.71) constitutes a tridiagonallinear system for the unknown eP'J+I, which
can be solved with minimal computational effort as discussed in the Appendix.
Although the Crank-Nicolson approximation to the one -dirnensional diffusion