In this chapter, we will prove convergence of the vorticity redistribution method for the Stokes or heat equation (3.21). The redistribution fractions are assumed to satisfy the redistribution equations (4.8) and following, to satisfy the positivity constraint (4.13), and to be restricted to vortices within a mutual distance (4.2), with . Since we are enforcing consistency in the norm, using the Fourier transform, while we have stability in the norm, and the redistribution fractions are only partly determined, the conventional convergence arguments need some modifications.
We will show convergence in the
norm by showing convergence of the Fourier transform of the numerical solution,
The total error consists of the error induced by discretizing the
initial data and the error induced by the redistribution method itself:
The first error due to discretizing the initial data
can be important if the initial data have only limited smoothness or if a
low-order smoothing function is used.
It depends on how the initial discretization is performed.
Typically the initial vortices are given a uniform spacing
and the initial vortex strength is taken
as
.
Since the initial vorticity field is evaluated only at the vortices,
some information is lost; aliasing makes indistinguishable
from the Fourier interpolant through the vorticity
values.
The total error due to discretization of the initial data can be written:
(5.4) |
The magnitude of the first of these two errors depends on the number of square integrable derivatives of the initial vorticity. It may be shown that if derivatives are square integrable, this error is of order ([219] pp. 198-206). In two dimensions has to be greater than one, but fractional values are allowed.
The second error is due to the vortex core. Assuming to be bounded, for nonzero times the order of this error is simply the order of accuracy of the vortex core. Thus, if the core is accurate , the overall accuracy of the computation is not affected by the core.
It follows that
for sufficiently accurate smoothing function and smooth initial data,
the only important error will be that due to the redistribution process.
To estimate this error,
we first define the local error in the Fourier transform
at time-level
to be the difference between the redistribution solution and the
exactly diffused solution from the previous time-step:
To estimate
, recall from chapter 4 that
the redistribution equations (4.8) and following
ensure vanishing of the first few powers of in the
error in (5.5).
The Taylor series remainder theorem can be used
to express the remaining difference
.
That expression is shown in Appendix A; it can be bounded as
(5.8) |
In this work, we will assume that , the absolute circulation of
the discretized initial data, is finite.
Note that this is a restriction
on the norm of the initial discrete vortex strengths, rather than
on the norm of the initial vorticity distribution.
However, the Cauchy inequality applied to
The final error in the vorticity is found from square
integration of (5.9) over all wavenumbers.
Thus the error due to redistribution is found to be:
To minimize this error, a relatively large core size is desirable. If we take the core size proportional to some small power of , the error will be . Since we can take as any positive number, we can obtain any order of accuracy arbitrarily close to . Note however that for a core with a finite order of accuracy, the first error in (5.3), due to discretizing the initial data, limits the maximum size of . We will discuss using a smoothing function to evaluate the vorticity further in the next chapter.
This completes the discussion of convergence for the Stokes equations. It is interesting to note that the true stability conditions are that and are bounded. Next we need to address how to find the redistribution fractions in an actual application of the scheme. We will address this in the next chapter on the numerical implementation of the convection and diffusion steps described in section 3.2.