136
3. Beyond the One-Way Wave Equation
cannot be used to show stability, since IIAI 11 > 1. In fact, numcrical calculations
show that the magnitude of the largest cigenvalue of A2AI is greatcr than unity
for alilbl, lei> 0, so the composite scheme is unconditionally unstable.
3.4 Diffusion, Sourees, and Sinks
Some background dissipation is present in most physical systems, and even when
the phenomena of interest are govemed by essentially inviscid processes, it is
often necessary to incorporate the effects of this dissipation in a numerical model.
Sources and sinks also may need to be included. In the following, we will consider
finite-difference methods for the incorporation of diffusive and Rayleigh-damping
processes. Numerical approximations to the pure diffusion equation
(3.68)
(where M > 0 is a molecular diffusivity) will be examined before considering the
combined effects of advection and diffusion or Rayleigh damping .
3.4.1 Pure Diffusion
Suppose that (3.68) is approximated using the forward-time centered-space scheme
The standard Von Neumann stability analysis yields an amplification factor for
this scheme of
Ak = 1 - 2v(1 - coskßx) ,
(3.69)
where v = M ßt / (ßx)2. The amplification factor is maximized for k = 1{ / ßX
(the 2ßX mode), and thus IAkl S 1 and the numerical solution decays with time,
provided that 0 S v s !. Note that in contrast to the conditional stability criteria obtained for finite-difference approximations to the advection equation, this
scheme does not remain stable as ßt, Sx ---+ 0 unless ßt decreases much more
rapidly than S», i.e., unless ßt / ßX < O(ßx). This makes the preceding scheme
very inefficient at high spatial resolution.
Although it guarantees that the solution will not blow up, the criterion 0 S v s
! is not adequate to ensure a qualitatively correct simulation of the 2ßX mode.
The amplitude bk of the kth Fourier mode of the exact solution to (3.68) satisfies
(3.70)
3. Beyond the One-Way Wave Equation
cannot be used to show stability, since IIAI 11 > 1. In fact, numcrical calculations
show that the magnitude of the largest cigenvalue of A2AI is greatcr than unity
for alilbl, lei> 0, so the composite scheme is unconditionally unstable.
3.4 Diffusion, Sourees, and Sinks
Some background dissipation is present in most physical systems, and even when
the phenomena of interest are govemed by essentially inviscid processes, it is
often necessary to incorporate the effects of this dissipation in a numerical model.
Sources and sinks also may need to be included. In the following, we will consider
finite-difference methods for the incorporation of diffusive and Rayleigh-damping
processes. Numerical approximations to the pure diffusion equation
(3.68)
(where M > 0 is a molecular diffusivity) will be examined before considering the
combined effects of advection and diffusion or Rayleigh damping .
3.4.1 Pure Diffusion
Suppose that (3.68) is approximated using the forward-time centered-space scheme
The standard Von Neumann stability analysis yields an amplification factor for
this scheme of
Ak = 1 - 2v(1 - coskßx) ,
(3.69)
where v = M ßt / (ßx)2. The amplification factor is maximized for k = 1{ / ßX
(the 2ßX mode), and thus IAkl S 1 and the numerical solution decays with time,
provided that 0 S v s !. Note that in contrast to the conditional stability criteria obtained for finite-difference approximations to the advection equation, this
scheme does not remain stable as ßt, Sx ---+ 0 unless ßt decreases much more
rapidly than S», i.e., unless ßt / ßX < O(ßx). This makes the preceding scheme
very inefficient at high spatial resolution.
Although it guarantees that the solution will not blow up, the criterion 0 S v s
! is not adequate to ensure a qualitatively correct simulation of the 2ßX mode.
The amplitude bk of the kth Fourier mode of the exact solution to (3.68) satisfies
(3.70)