Manuscript En02-E@2 · En02-E-v2.tex · section · c2769986

Radial boundary-error estimate

Radial boundary-error estimate · lines 1460–1482

For (eq:radial),

v = \frac{ℓ ( ℓ + 1 )}{r _{h}} + \frac{1}{2 r _{h}} .

If $W = V 1_{[x_{a}, x_{b}]}$ , then $w = v - d$ , and

d = ℓ (ℓ + 1) (\frac{1}{r _{h}} - \frac{1}{r _{a}}) + \frac{1}{2 r _{h}} - \frac{r _{h}}{2 r _{a} ^{2}} + \frac{ℓ ( ℓ + 1 )}{r _{b}} + \frac{r _{h}}{2 r _{b} ^{2}},

where $r_{a} = r (x_{a})$ , $r_{b} = r (x_{b})$ . These are exact potential-tail integrals, not ODE solver errors.

The implementation computes the near-horizon potential using $z = r - 1$ , rather than subtracting a rounded value of $r$ from one. This avoids prematurely rounding the exponentially small near-horizon potential to zero at the pilot boundaries.

Displayed formulas in this passage

unlabeled · line 1463
$v = \frac{ℓ ( ℓ + 1 )}{r _{h}} + \frac{1}{2 r _{h}} .$
unlabeled · line 1467
$d = ℓ (ℓ + 1) (\frac{1}{r _{h}} - \frac{1}{r _{a}}) + \frac{1}{2 r _{h}} - \frac{r _{h}}{2 r _{a} ^{2}} + \frac{ℓ ( ℓ + 1 )}{r _{b}} + \frac{r _{h}}{2 r _{b} ^{2}},$

Radial boundary-error estimate

Displayed formulas in this passage

Nearby in this section