A. Detailed derivations

This appendix gives some derivations mentioned earlier in the various chapters. First is the derivation of the redistribution equations (4.8) and following. We write the difference between the Fourier transform of the redistributed vorticity (4.5) and the exactly diffused vorticity (4.6) in terms of the viscous scale $h_v=\sqrt{\nu\Delta t}$ and the scaled relative positions (4.7):

$$\widehat\omega^{n+1} - \widehat\omega^{n+1}_e = \widehat\phi... ...rm i} h_v\vec k\cdot\vec\xi_{ij}} - e^{- h_v^2 k^2} \right)\ .$$ (A.1)

In the redistribution method, this error is made small of order $O(h_v k)^{M+2}$, or $O(\Delta t)^{(M+2)/2}$ for any finite wavenumber $k$, by equating the Taylor series expansions of the two terms within the parentheses to that order. This produces, for any $m\le M+1$,

$$\displaystyle \sum_j f^n_{ij} \left(k_1 \xi_{1ij} + k_2 \xi_{2ij}\right)^m = 0 \quad \mbox{($m$\ odd)}\ ,$$

$$= \displaystyle \frac{m!}{(\frac12 m)!} (k_1^2 + k_2^2)^{\frac12 m} \quad \mbox{($m$\ even)}\ .$$ (A.2)

Expanding using the binomial theorem, the individual equations become, for $m_1+m_2\le M+1$,

$$\displaystyle \sum_j f^n_{ij} \xi_{1ij}^{m_1} \xi_{2ij}^{m_2} = 0 \ \mbox{($m_1$\ or $m_2$\ odd)} \ ,$$

$$= \displaystyle \frac{m_1! m_2!}{(\frac12 m_1)!(\frac12 m_2)!} \ \mbox{($m_1$\ and $m_2$\ even)}\ .$$ (A.3)

These are the redistribution equations written out in (4.8) and following.

The remaining error in the Fourier transform, needed in chapter 5, is according to the Taylor series remainder theorem

$$\widehat\omega^{n+1} - \widehat\omega^{n+1}_e = \widehat\phi(k \delta) \sum_i \Gamma_i^n e^{-{\rm i}\vec k\cdot\vec x_i} \times$$

$$\qquad \Bigg\{ \frac{(k h_v R)^{M+2}}{(M+2)!} \sum_j f^n_{ij} (\c... ...cos\beta_{ij\vec k}) \left(\frac{\vec k \cdot\vec \xi_{ij}}{k R}\right)^{M+2} -$$

$$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{(- k^2 h_v^2)^{\frac12 M_e+1}}{(\frac12 M_e+1)!} e^{-\gamma^2_k} \Bigg\}\ ,$$ (A.4)

where the values of $\alpha_{ij\vec k}$, $\beta_{ij\vec k}$ and $\gamma_k$ represent the undetermined midpoints in the remainder theorem, and $M_e$ is the even integer $M$ or $M+1$.

To find the lower bound to the redistribution radius mentioned in section 6.2.3, we integrate (A.2) over the unit circle to produce

$$\sum_j f^n_{ij} \xi_{ij}^m = 2^m ({\textstyle\frac12}m)! \qquad \mbox{($m\le M+1$\ and $m$\ even)} \ .$$ (A.5)

Hence, in terms of the even integer $M_e=M$ or $M+1$,

$$\max_j \xi_{ij}^2 \ge \sum_j f^n_{ij} \xi_{ij}^{M_e} \bigg/ \sum_j f^n_{ij} \xi_{ij}^{M_e-2} = 2 M_e\ ,$$ (A.6)

which implies the lower bound for the redistribution radius.

Up to fourth-order accuracy, this estimate for the minimum radius is precise. It may be verified by direct substitution into the redistribution equations that a positive second-order solution is obtained by spreading the fractions evenly over the circle with scaled radius $R=2$. Similarly, a positive fourth-order solution is obtained by giving the vortex being redistributed a fraction $0.5$ and spreading the other half evenly over the circle $R=\sqrt{8}$.

Next we verify an assertion made in section 6.2.3: as long as all vortices are redistributed, the region containing the vortices must expand a finite scaled amount in each direction. To do so, we derive a lower bound $\xi_{1\hbox{max}}$ to $\max_j(\xi_{1ij})$ using the cases $m_2=0$ and $m_1=0$, 1, and 2 of (A.3):

$$8 = \sum_j f^n_{ij} (\vert\xi_{1ij}\vert+\xi_{1ij})^2 + \sum_j f^n_{ij} (\vert\xi_{1ij}\vert-\xi_{1ij})^2$$

$$\le 4 \xi_{1\hbox{max}}^2 + \sum_j f^n_{ij} (\vert\xi_{1ij}\vert-\xi_{1ij}) 2R$$

$$= 4 \xi_{1\hbox{max}}^2 + \sum_j f^n_{ij} (\vert\xi_{1ij}\vert+\xi_{1ij}) 2R$$

$$\le 4 \xi_{1\hbox{max}}^2 + 4 \xi_{1\hbox{max}} R\ .$$ (A.7)

The solution of the quadratic shows that

$$\xi_{1\hbox{max}}\ge 4/(R+\sqrt{R^2+8}) \ .$$ (A.8)

Applied to the vortex at the largest value of $x$, this value describes how much the vortex distribution needs to expand in the $x$-direction in order for the redistribution equations to be solvable at that vortex. Since the redistribution problem is independent of the angular position of the coordinate system, this minimum expansion applies in any direction.

Next we verify an assertion made in section 6.2.3 and subsection 9.5: for third-order accuracy or higher, the scaled spacing between the vortices cannot be arbitrarily large. Defining $\xi_{\hbox{min}}=\min_{j\ne i}\{\xi_{ij}\}$ and using (A.5) for $m=4$ and $m=2$, it is seen that

$$32 = \sum_j f^n_{ij} \xi_{ij}^4 \ge \sum_j f^n_{ij} \xi_{ij}^2 \xi_{\hbox{min}}^2 = 4 \xi_{\hbox{min}}^2 \ .$$ (A.9)

Hence the vortices cannot be spaced further apart than a scaled distance $\sqrt{8}$.

Finally we verify an assertion made in section 6.2.3: there are finite values $R$ and $d$ so that a positive solution to the redistribution equations exists within the circle with scaled radius $R$ provided that there are no square holes exceeding a scaled size $d$ in the distribution of the vortices. To do so, we first note that the diffusing delta function

$$f(\xi_{1i},\xi_{2i}) = \frac{1}{4\pi} e^{-(\xi_{1i}^2+\xi_{2i}^2)/4}$$ (A.10)

gives an exact continuous solution to our redistribution equations, replacing $\sum_j$ by $\int\int {\rm d}\xi_{1i} {\rm d}\xi_{2i}$. We now discretize this continuous solution using a cut-off at radius $R$ and a subdivision of the remaining domain into squares of size $d$. We then select one vortex in each square and give it fraction

$$f^n_{ij} = d^2 f(\xi_{1ij},\xi_{2ij}) + d^2 \sum_{m,n} c_{mn} p_m(\xi_{1ij}) p_n(\xi_{2ij})\ ,$$ (A.11)

where the $p_n(\xi)$ are for $\vert\xi\vert\le 1$ polynomials of degree $M+1$ satisfying

$$\int_{-1}^1 p_n(\xi) \xi^m {\rm d}\xi = \delta_{mn} \qquad (m=0, \ldots, M+1)$$ (A.12)

and zero elsewhere. In the above expression for $f^n_{ij}$, the first term provides a positive approximate solution to the redistribution equations, since the sums over $j$ become straightforward numerical approximations to the corresponding integrals of the continuous function $f$. The second term gives corrections that make this approximation exact for suitable values of the constants $c_{mn}$. When $R$ is large enough and $d$ is small enough, these corrections do not change the positivity of the first term. This can be seen as follows: since the polynomials are bounded, there is a finite maximum value for the constants $\vert c_{mn}\vert$ below which the correction terms cannot change the sign of the first term to $f^n_{ij}$. Further, since the redistribution equations give a system of equations for the $c_{mn}$ that tends to a unit matrix, there is a value of $d$ below which the maximum $\vert c_{mn}\vert$ can be bounded by a multiple of the maximum error due to the first term in (A.11). That error can be reduced to any finite amount by selecting a large enough $R$ and a small enough $d$ to make the numerical integrals sufficiently accurate. Hence the required positive total solution (A.11) can always be assured for some finite $R$ and $d$.

At least for the case of first-order accuracy, $M=1$, for any $R$ greater than the minimum value $R=2$, a finite hole size $d$ exists that ensures a positive solution. This can be seen by selecting nine vortices to satisfy the redistribution equations. Eight of these are chosen as closely as possible to eight equally spaced points on the outside circle and given a nominal fraction $f^n_{ij}=1/2R^2$, and the last point is chosen to be the vortex being redistributed, and given a nominal weight $1-4/R^2$. This satisfies the redistribution equations approximately, and it is readily seen that for these nominal positions, the needed corrections in the weights to make the approximation exact can be bounded by the errors. Thus, similar as in the derivation above, the corrections do not change the sign of the weights when $d$ is small enough. The actual value of $d$ is unknown, but clearly $d$ must tend to zero when $R\to 2$; the allowed hole size must be small enough to ensure that there are vortices outside the circle $R=2$ within which no solution exists.