Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2

Better Interface for 51BiPolytrope Stability Study

This is Part 2 of an extended chapter discussion. For Part 1, go here.

Discretize for Numerical Integration (continued)

General Discretization

Fourth Approximation

Let's assume that we know the four quantities, $x_{J-1},x_J,(x_J{)}^{\text{'}}\equiv (dx/d\tilde{r}{)}_J$, and $(x_{J-1}{)}^{\text{'}}\equiv (dx/d\tilde{r}{)}_{J-1}$ and want to project forward to determine, $x_{J+1}$. We should assume that, locally, the displacement function $x$ is cubic in $\tilde{r}$, that is,

$x$	$=$	$a+b\tilde{r}+c{\tilde{r}}^2+e{\tilde{r}}^3$
$\Rightarrow \frac{dx}{d\tilde{r}}$	$=$	$b+2c\tilde{r}+3e{\tilde{r}}^2,$

where we have four unknowns, $a,b,c,e$. These can be determined by appropriately combining the four relations,

$(x_J{)}^{\text{'}}$	$=$	$b+2c{\tilde{r}}_J+3e{\tilde{r}}_J^2,$
$(x_{J-1}{)}^{\text{'}}$	$=$	$b+2c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r})+3e({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2,$
$x_J$	$=$	$a+b{\tilde{r}}_J+c{\tilde{r}}_J^2+e{\tilde{r}}_J^3,$
$x_{J-1}$	$=$	$a+b({\tilde{r}}_J-\mathrm{\Delta }\tilde{r})+c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2+e({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^3,$

The difference between the first two expressions gives,

$(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}$	$=$	$[2c{\tilde{r}}_J+3e{\tilde{r}}_J^2]-[2c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r})+3e({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2]$
	$=$	$2c{\tilde{r}}_J+3e{\tilde{r}}_J^2-[2c{\tilde{r}}_J-2c\mathrm{\Delta }\tilde{r}+3e({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$
	$=$	$2c\mathrm{\Delta }\tilde{r}+6e{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3e\mathrm{\Delta }{\tilde{r}}^2$
$\Rightarrow 2c\mathrm{\Delta }\tilde{r}$	$=$	$[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+3e\mathrm{\Delta }{\tilde{r}}^2-6e{\tilde{r}}_J\mathrm{\Delta }\tilde{r}$
$\Rightarrow c$	$=$	$\left[\frac{(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}}{2\mathrm{\Delta }\tilde{r}}\right]+3e[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J].$

And the difference between the last two expressions gives,

$x_J-x_{J-1}$	$=$	$[b{\tilde{r}}_J+c{\tilde{r}}_J^2+e{\tilde{r}}_J^3]-[b({\tilde{r}}_J-\mathrm{\Delta }\tilde{r})+c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2+e({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^3]$
	$=$	$b\mathrm{\Delta }\tilde{r}+c(2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2)+e{\tilde{r}}_J^3-e({\tilde{r}}_J-\mathrm{\Delta }\tilde{r})({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)$
	$=$	$b\mathrm{\Delta }\tilde{r}+c(2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2)+e{\tilde{r}}_J^3-e[({\tilde{r}}_J)({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)-(\mathrm{\Delta }\tilde{r})({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$
	$=$	$b\mathrm{\Delta }\tilde{r}+c(2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2)-e[-3{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}+3{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^2-\mathrm{\Delta }{\tilde{r}}^3]$
	$=$	$b\mathrm{\Delta }\tilde{r}+c(2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2)+e[3{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^2+\mathrm{\Delta }{\tilde{r}}^3]$
$\Rightarrow b\mathrm{\Delta }\tilde{r}$	$=$	$[x_J-x_{J-1}]+2c\mathrm{\Delta }\tilde{r}[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]-e[3{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^2+\mathrm{\Delta }{\tilde{r}}^3]$
	$=$	$[x_J-x_{J-1}]+\left\{\right[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+3e\mathrm{\Delta }\tilde{r}[\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J\left]\right\}[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]-3e\mathrm{\Delta }\tilde{r}[{\tilde{r}}_J^2-{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{3}]$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+3e\mathrm{\Delta }\tilde{r}[\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}-{\tilde{r}}_J\mathrm{\Delta }\tilde{r}]-3e\mathrm{\Delta }\tilde{r}[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J^2]-3e\mathrm{\Delta }\tilde{r}[{\tilde{r}}_J^2-{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{3}]$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]-3e\mathrm{\Delta }\tilde{r}\left\{\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}]+[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J^2]+[{\tilde{r}}_J^2-{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{3}\left]\right\}$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}].$

Summary #1: In terms of the coefficient, $e$ … $b\mathrm{\Delta }\tilde{r}$ $=$ $[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}],$ $2c\mathrm{\Delta }\tilde{r}$ $=$ $[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+e\mathrm{\Delta }\tilde{r}[3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J].$

Hence, from the first of the four relations, we find that,

$(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}-3e{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}$	$=$	$(b\mathrm{\Delta }\tilde{r})+(2c\mathrm{\Delta }\tilde{r}){\tilde{r}}_J$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}]+\left\{\right[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+e\mathrm{\Delta }\tilde{r}[3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J\left]\right\}{\tilde{r}}_J$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\left[\frac{\mathrm{\Delta }\tilde{r}}{2}\right]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2]$
	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\left[\frac{\mathrm{\Delta }\tilde{r}}{2}\right]+e\mathrm{\Delta }\tilde{r}[-3{\tilde{r}}_J^2+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}]$
$\Rightarrow (x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}$	$=$	$[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\left[\frac{\mathrm{\Delta }\tilde{r}}{2}\right]+e\left[\frac{\mathrm{\Delta }{\tilde{r}}^3}{2}\right]$
$\Rightarrow e\left[\frac{\mathrm{\Delta }{\tilde{r}}^3}{2}\right]$	$=$	$-[x_J-x_{J-1}]-[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\left[\frac{\mathrm{\Delta }\tilde{r}}{2}\right]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}$
	$=$	$[x_{J-1}-x_J]+[(x_{J-1}{)}^{\text{'}}+(x_J{)}^{\text{'}}]\frac{\mathrm{\Delta }\tilde{r}}{2}$
$\Rightarrow e\mathrm{\Delta }{\tilde{r}}^3$	$=$	$2[x_{J-1}-x_J]+[(x_{J-1}{)}^{\text{'}}+(x_J{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}.$

Finally, from the third of the four relations, we can evaluate the coefficient, $a$; specifically,

$x_J-a-e{\tilde{r}}_J^3$	$=$	$b{\tilde{r}}_J+c{\tilde{r}}_J^2$
	$=$	$\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\left\{b\mathrm{\Delta }\tilde{r}\right\}+\frac{{\tilde{r}}_J^2}{2\mathrm{\Delta }\tilde{r}}\left\{2c\mathrm{\Delta }\tilde{r}\right\}$
	$=$	$\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\left\{\right[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}\left]\right[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }\tilde{r}[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}\left]\right\}+\frac{{\tilde{r}}_J^2}{2\mathrm{\Delta }\tilde{r}}\left\{\right[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+e\mathrm{\Delta }\tilde{r}[3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J\left]\right\}$
	$=$	$\left\{\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right[x_J-x_{J-1}]+\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}\left]\right[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e{\tilde{r}}_J[3{\tilde{r}}_J^2-3{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\frac{\mathrm{\Delta }{\tilde{r}}^2}{2}\left]\right\}+\left\{\frac{{\tilde{r}}_J^2}{2\mathrm{\Delta }\tilde{r}}\right[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]+\frac{e{\tilde{r}}_J^2}{2}[3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J\left]\right\}$
	$=$	$\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{{\tilde{r}}_J}{2}-\frac{{\tilde{r}}_J^2}{2\mathrm{\Delta }\tilde{r}}]+e[3{\tilde{r}}_J^3-3{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}+\frac{{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^2}{2}]+e[\frac{3{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}}{2}-3{\tilde{r}}_J^3]$
	$=$	$\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]+\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]+\frac{e\mathrm{\Delta }\tilde{r}}{2}[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2].$

That is,

$a$	$=$	$x_J-\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]-\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]-e\left\{\frac{\mathrm{\Delta }\tilde{r}}{2}\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2]+{\tilde{r}}_J^3\}$
	$=$	$x_J-\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]-\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]-e\mathrm{\Delta }{\tilde{r}}^3\left\{\frac{1}{2\mathrm{\Delta }{\tilde{r}}^2}\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2]+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\}$

Summary #2: In terms of the coefficient, $e$ … $a$ $=$ $x_J-\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]-\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]-e\mathrm{\Delta }{\tilde{r}}^3\left\{\frac{1}{2\mathrm{\Delta }{\tilde{r}}^2}\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2]+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\},$ $b\mathrm{\Delta }\tilde{r}$ $=$ $[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3{\tilde{r}}_J^2}{\mathrm{\Delta }{\tilde{r}}^2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+\frac{1}{2}],$ $c\mathrm{\Delta }{\tilde{r}}^2$ $=$ $[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\frac{\mathrm{\Delta }\tilde{r}}{2}+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3}{2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}],$ $e\mathrm{\Delta }{\tilde{r}}^3$ $=$ $2[x_{J-1}-x_J]+[(x_{J-1}{)}^{\text{'}}+(x_J{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}.$

This is test ... ${\tilde{r}}_J={\tilde{r}}_i+\mathrm{\Delta }\tilde{r}$ $\mathrm{\Delta }\tilde{r}$ $x_J$ $x_{J-1}$ $(x_J{)}^{\text{'}}$ $(x_{J-1}{)}^{\text{'}}$ 0.01740039 0.001936393 -4.695376 -4.547832 -116.0119 -76.19513

$a$ $=$ $-3.36955-2.76645-e\mathrm{\Delta }{\tilde{r}}^3(608.9698)=-232.7874,$ $b\mathrm{\Delta }\tilde{r}$ $=$ $0.5067329+e\mathrm{\Delta }{\tilde{r}}^3(215.7856)=+80.819698,$ $c\mathrm{\Delta }{\tilde{r}}^2$ $=$ $-0.0385505+e\mathrm{\Delta }{\tilde{r}}^3(-25.45794)=-9.51370,$ $e\mathrm{\Delta }{\tilde{r}}^3$ $=$ $0.3721883.$ Hence, $x_J$ $=$ $a+b\mathrm{\Delta }\tilde{r}\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right)+c\mathrm{\Delta }{\tilde{r}}^2(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+e\mathrm{\Delta }{\tilde{r}}^3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3$   $=$ $-232.7874+726.2442-768.2108+270.0593=-4.68369.$ Higher precision value (from Excel) is $x_J=-4.695376,$ which precisely matches the input value. Also from Excel, $x_{J-1}=-4.547832$ and $x_{J+1}=-3.803455.$

As a result,

$x_{J+1}$	$=$	$\left\{a\right\}+({\tilde{r}}_J+\mathrm{\Delta }\tilde{r})\left\{b\right\}+({\tilde{r}}_J+\mathrm{\Delta }\tilde{r}{)}^2\{c\}+({\tilde{r}}_J+\mathrm{\Delta }\tilde{r}{)}^3\left\{e\right\}$
	$=$	$\left\{a\right\}+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)\left\{b\mathrm{\Delta }\tilde{r}\right\}+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1{)}^2\{c\mathrm{\Delta }{\tilde{r}}^2\}+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1{)}^3\left\{e\mathrm{\Delta }{\tilde{r}}^3\right\}$
	$=$	$x_J-\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]-\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]-e\mathrm{\Delta }{\tilde{r}}^3\left\{\frac{1}{2\mathrm{\Delta }{\tilde{r}}^2}\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2]+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\}$
		$+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)\left\{\right[x_J-x_{J-1}]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}\left]\right[\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J]+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3{\tilde{r}}_J^2}{\mathrm{\Delta }{\tilde{r}}^2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+\frac{1}{2}\left]\right\}$
		$+\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1\left]\right\{[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\frac{\mathrm{\Delta }\tilde{r}}{2}+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3}{2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}]\}$
		$+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1\left]\right\{e\mathrm{\Delta }{\tilde{r}}^3\}$

$\Rightarrow x_{J+1}$	$=$	$x_J-\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}[x_J-x_{J-1}]-\frac{1}{2\mathrm{\Delta }\tilde{r}}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-{\tilde{r}}_J^2]+[x_J-x_{J-1}](\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\frac{\mathrm{\Delta }\tilde{r}}{2}-{\tilde{r}}_J](\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1\left]\right[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\frac{\mathrm{\Delta }\tilde{r}}{2}$
		$+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3{\tilde{r}}_J^2}{\mathrm{\Delta }{\tilde{r}}^2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+\frac{1}{2}](\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+e\mathrm{\Delta }{\tilde{r}}^3[\frac{3}{2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}]\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]+e\mathrm{\Delta }{\tilde{r}}^3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1\left)\right[(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]-e\mathrm{\Delta }{\tilde{r}}^3\left\{\frac{1}{2\mathrm{\Delta }{\tilde{r}}^2}\right[{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2]+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\}$
	$=$	$2x_J-x_{J-1}-\frac{1}{2}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][{\tilde{r}}_J-\frac{{\tilde{r}}_J^2}{\mathrm{\Delta }\tilde{r}}]+\frac{1}{2}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}][\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J](\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+[\frac{{\tilde{r}}_J^2}{\mathrm{\Delta }\tilde{r}}+2{\tilde{r}}_J+\mathrm{\Delta }\tilde{r}][(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\frac{1}{2}$
		$+e\mathrm{\Delta }{\tilde{r}}^3\left\{\right[\frac{3{\tilde{r}}_J^2}{\mathrm{\Delta }{\tilde{r}}^2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+\frac{1}{2}\left]\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+[\frac{3}{2}-\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\left]\right[(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]-[\frac{1}{2\mathrm{\Delta }{\tilde{r}}^2}({\tilde{r}}_J\mathrm{\Delta }\tilde{r}-3{\tilde{r}}_J^2)+\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\right]\}$
	$=$	$2x_J-x_{J-1}+\frac{1}{2}[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\left\{\right[\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J\left]\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1)+[\frac{{\tilde{r}}_J^2}{\mathrm{\Delta }\tilde{r}}+2{\tilde{r}}_J+\mathrm{\Delta }\tilde{r}]-[{\tilde{r}}_J-\frac{{\tilde{r}}_J^2}{\mathrm{\Delta }\tilde{r}}\left]\right\}$
		$+e\mathrm{\Delta }{\tilde{r}}^3\left\{\right[3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3-3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{1}{2}\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right)]+[3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2-3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}})+\frac{1}{2}]+\frac{3}{2}\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]-3[(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3+2(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right)]$
		$+\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3+2(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\left)\right]+\left[\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]+[-\frac{{\tilde{r}}_J}{2\mathrm{\Delta }\tilde{r}}+\frac{3}{2}(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2-(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^3\left]\right\}$

Continuing …

$x_{J+1}$	$=$	$2x_J-x_{J-1}+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}$
		$+e\mathrm{\Delta }{\tilde{r}}^3\left\{\right[\frac{{\tilde{r}}_J}{2\mathrm{\Delta }\tilde{r}}]+[-3\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right)+\frac{1}{2}]+\frac{3}{2}[(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]-\left[6\right(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+3\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}\right)]$
		$+3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+[\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+1]+[\frac{{\tilde{r}}_J}{2\mathrm{\Delta }\tilde{r}}+\frac{3}{2}\left(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2\right]\}$
	$=$	$2x_J-x_{J-1}+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}$
		$+e\mathrm{\Delta }{\tilde{r}}^3\{\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}-6(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}})+3+\frac{3{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}-6(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2+\frac{2{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}+3(\frac{{\tilde{r}}_J}{\mathrm{\Delta }\tilde{r}}{)}^2\}$
	$=$	$2x_J-x_{J-1}+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}+3e\mathrm{\Delta }{\tilde{r}}^3$

Finally we may write,

$x_{J+1}$	$=$	$2x_J-x_{J-1}+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}+3\left\{2\right[x_{J-1}-x_J]+[(x_{J-1}{)}^{\text{'}}+(x_J{)}^{\text{'}}\left]\mathrm{\Delta }\tilde{r}\right\}$
	$=$	$2x_J-x_{J-1}+6[x_{J-1}-x_J]+[(x_J{)}^{\text{'}}-(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}+3[(x_{J-1}{)}^{\text{'}}+(x_J{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}$
	$=$	$[5x_{J-1}-4x_J]+[4(x_J{)}^{\text{'}}+2(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}.$

This is test ... $\mathrm{\Delta }\tilde{r}$ $x_J$ $x_{J-1}$ $(x_J{)}^{\text{'}}$ $(x_{J-1}{)}^{\text{'}}$ 0.001936393 -4.695376 -4.547832 -116.0119 -76.19513

$x_{J+1}$ $=$ $[5x_{J-1}-4x_J]+[4(x_J{)}^{\text{'}}+2(x_{J-1}{)}^{\text{'}}]\mathrm{\Delta }\tilde{r}=-5.15132.$

Fifth Approximation

Let's assume that we know the three quantities, $x_{J-1},x_J,(x_J{)}^{\text{'}}\equiv (dx/d\tilde{r}{)}_J$, and want to project forward to determine, $x_{J+1}$. Here we will assume that, locally, the displacement function $x$ has only an even-power dependence on $\tilde{r}$, that is,

$x$	$=$	$a+b{\tilde{r}}^2+c{\tilde{r}}^4$
$\Rightarrow \frac{dx}{d\tilde{r}}$	$=$	$2b\tilde{r}+4c{\tilde{r}}^3,$

where we have three unknowns, $a,b,c$. These can be determined by appropriately combining the three relations,

$(x_J{)}^{\text{'}}$	$=$	$2b{\tilde{r}}_J+4c{\tilde{r}}_J^3,$
$x_J$	$=$	$a+b{\tilde{r}}_J^2+c{\tilde{r}}_J^4,$
$x_{J-1}$	$=$	$a+b({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2+c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^4,$

The difference between the last two expressions gives,

$x_J-x_{J-1}$	$=$	$[b{\tilde{r}}_J^2+c{\tilde{r}}_J^4]-[b({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2+c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^4]$
	$=$	$b{\tilde{r}}_J^2+c{\tilde{r}}_J^4-b({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)-c({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)$
	$=$	$c{\tilde{r}}_J^4+2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-b\mathrm{\Delta }{\tilde{r}}^2-c[{\tilde{r}}_J^2({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+\mathrm{\Delta }{\tilde{r}}^2({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$
	$=$	$c{\tilde{r}}_J^4+2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-b\mathrm{\Delta }{\tilde{r}}^2-c[{\tilde{r}}_J^4-2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+{\tilde{r}}^2\mathrm{\Delta }{\tilde{r}}^2-2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+4{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3+{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3+\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$b[2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4].$

Repeat, to check … $x_J-x_{J-1}$ $=$ $[b{\tilde{r}}_J^2+c{\tilde{r}}_J^4]-[b({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^2+c({\tilde{r}}_J-\mathrm{\Delta }\tilde{r}{)}^4]$   $=$ $[b{\tilde{r}}_J^2+c{\tilde{r}}_J^4]-[b({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+c({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$   $=$ $[c{\tilde{r}}_J^4]+b[2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2]-c[{\tilde{r}}_J^2({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+\mathrm{\Delta }{\tilde{r}}^2({\tilde{r}}_J^2-2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$   $=$ $b[2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}-\mathrm{\Delta }{\tilde{r}}^2]-c[(-2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2)+(-2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+4{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3)+({\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3+\mathrm{\Delta }{\tilde{r}}^4)]$   $=$ $2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$ Hence, $x_J-x_{J-1}$ $=$ $[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}-4c{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$   $=$ $[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}\left]\right[\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}-1]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$ $\Rightarrow x_J-x_{J-1}-(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)]$ $=$ $c[2{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$   $=$ $c[-4{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$ $\Rightarrow 4c{\tilde{r}}_J^4\cdot {𝒜}$ $=$ $x_{J-1}-x_J+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)]$   $=$ $x_{J-1}-x_J+(x_J{)}^{\text{'}}{\tilde{r}}_J[\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]$ where, ${𝒜}$ $\equiv$ $\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4].$ Also, $2b{\tilde{r}}_J^2$ $=$ $(x_J{)}^{\text{'}}{\tilde{r}}_J-4c{\tilde{r}}_J^4$ $\Rightarrow 2b{\tilde{r}}_J^2\cdot {𝒜}$ $=$ $(x_J{)}^{\text{'}}{\tilde{r}}_J\cdot {𝒜}-\{x_{J-1}-x_J+(x_J{)}^{\text{'}}{\tilde{r}}_J[\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]\}$   $=$ $x_J-x_{J-1}+(x_J{)}^{\text{'}}{\tilde{r}}_J[{𝒜}-\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right].$

From the first expression, we also see that,

$2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}$

$=$

$(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}-4c{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}.$

Therefore we have,

$x_J-x_{J-1}$	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}-4c{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+[2c{\tilde{r}}_J^2\mathrm{\Delta }\tilde{r}\left]\right[\mathrm{\Delta }\tilde{r}-2{\tilde{r}}_J]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[2{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2-4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}]+c[4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}-6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]+c[-4{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3-\mathrm{\Delta }{\tilde{r}}^4]$
$\Rightarrow c\mathrm{\Delta }{\tilde{r}}^2[4{\tilde{r}}_J^2-4{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2]$	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}$
$\Rightarrow 4c{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2[1-\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]$	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}$
$\Rightarrow 4c{\tilde{r}}_J^4\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$	$=$	$[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}.$

Hence also,

$2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)$	$=$	$(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-\{4c{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2\}$
$\Rightarrow 2b{\tilde{r}}_J\mathrm{\Delta }\tilde{r}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)[1-\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]$	$=$	$(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\left)\right[1-\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]-\left\{\right[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}][1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}\}$
	$=$	$x_J-x_{J-1}+(x_J{)}^{\text{'}}(\frac{\mathrm{\Delta }{\tilde{r}}^2}{{\tilde{r}}_J}\left)\right[1-\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]-[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left]\right[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]$
	$=$	$x_J-x_{J-1}+(x_J{)}^{\text{'}}(\frac{\mathrm{\Delta }{\tilde{r}}^2}{{\tilde{r}}_J}\left)\right[1-\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]-(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}+\frac{1}{2}(x_J{)}^{\text{'}}\left(\frac{\mathrm{\Delta }{\tilde{r}}^2}{{\tilde{r}}_J}\right)$
$\Rightarrow 2b{\tilde{r}}_J^2\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$	$=$	$x_J-x_{J-1}+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]$

Finally,

$a$	$=$	$x_J-b{\tilde{r}}_J^2-c{\tilde{r}}_J^4$
$\Rightarrow a\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]-b{\tilde{r}}_J^2[(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]-c{\tilde{r}}_J^4\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
		$-\frac{1}{2}\{x_J-x_{J-1}+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]\}$
		$-\frac{1}{4}\left\{\right[(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}][1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}\}$
	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]-\frac{1}{2}\{x_J-x_{J-1}\}-\frac{1}{4}\{-x_J+x_{J-1}\}$
		$-\frac{1}{2}\{(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]\}-\frac{1}{4}\{(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)\left]\right\}$
	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+\frac{x_{J-1}-x_J}{4}$
		$+\{(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{2}-\frac{1}{4}+\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-\frac{3}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]\}$
	$=$	$\frac{x_{J-1}-x_J}{4}+x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right].$

Repeat, to check … $4[x_J-a]\cdot {𝒜}$ $=$ $4b{\tilde{r}}_J^2\cdot {𝒜}+4c{\tilde{r}}_J^4\cdot {𝒜}=2\{2b{\tilde{r}}_J^2\cdot {𝒜}\}+\{4c{\tilde{r}}_J^4\cdot {𝒜}\}$   $=$ $2\{x_J-x_{J-1}+(x_J{)}^{\text{'}}{\tilde{r}}_J[{𝒜}-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\left]\right\}+\{x_{J-1}-x_J+(x_J{)}^{\text{'}}{\tilde{r}}_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\left]\right\}$

Summary: $a\cdot {𝒜}$ $=$ $\frac{x_{J-1}-x_J}{4}+x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right],$ $2b{\tilde{r}}_J^2\cdot {𝒜}$ $=$ $x_J-x_{J-1}+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3],$ $\Rightarrow 4c{\tilde{r}}_J^4\cdot {𝒜}$ $=$ $(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1},$ where, ${𝒜}$ $\equiv$ $\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4].$

Let's now determine the expression for $x_{J+1}$. We begin by writing …

$x_{J+1}$	$=$	$a+b({\tilde{r}}_J+\mathrm{\Delta }\tilde{r}{)}^2+c({\tilde{r}}_J+\mathrm{\Delta }\tilde{r}{)}^4$
	$=$	$a+b({\tilde{r}}_J^2+2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+c[{\tilde{r}}_J^2({\tilde{r}}_J^2+2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}({\tilde{r}}_J^2+2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)+\mathrm{\Delta }{\tilde{r}}^2({\tilde{r}}_J^2+2{\tilde{r}}_J\mathrm{\Delta }\tilde{r}+\mathrm{\Delta }{\tilde{r}}^2)]$
	$=$	$a+b{\tilde{r}}_J^2[1+\frac{2\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]+c[({\tilde{r}}_J^4+2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2)+(2{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+4{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3)+({\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+2{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3+\mathrm{\Delta }{\tilde{r}}^4)]$
	$=$	$a+b{\tilde{r}}_J^2[1+\frac{2\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]+c[{\tilde{r}}_J^4+4{\tilde{r}}_J^3\mathrm{\Delta }\tilde{r}+6{\tilde{r}}_J^2\mathrm{\Delta }{\tilde{r}}^2+4{\tilde{r}}_J\mathrm{\Delta }{\tilde{r}}^3+\mathrm{\Delta }{\tilde{r}}^4]$
	$=$	$a+b{\tilde{r}}_J^2[1+2(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]+c{\tilde{r}}_J^4[1+4\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+6(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+4(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
	$=$	$a+2b{\tilde{r}}_J^2[\frac{1}{2}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]+4c{\tilde{r}}_J^4[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right].$

This means that,

${𝒜}\cdot x_{J+1}$	$=$	$a\cdot {𝒜}$
		$+2b{\tilde{r}}_J^2\cdot {𝒜}[\frac{1}{2}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]$
		$+4c{\tilde{r}}_J^4\cdot {𝒜}[\frac{1}{4}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
	$=$	$\frac{x_{J-1}-x_J}{4}+x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]$
		$+\{x_J-x_{J-1}+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]\left\}\right[\frac{1}{2}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]$
		$+\{(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}]-x_J+x_{J-1}\left\}\right[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
	$=$	$\frac{1}{4}{x}_{J-1}-\frac{1}{4}{x}_J+x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\right]$
		$+\{x_J-x_{J-1}\}[\frac{1}{2}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\left]\right[\frac{1}{2}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]$
		$+\{-x_J+x_{J-1}\}[\frac{1}{4}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[1-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}\left]\right[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$

Keep going …

${𝒜}\cdot x_{J+1}$	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
		$+\{x_J-x_{J-1}\}[-\frac{1}{4}+\frac{1}{2}+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J})+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{4}-\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3-\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\left]\right[\frac{1}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\left]\right[\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-1+\frac{3}{2}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\left]\right[\frac{1}{2}]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-\frac{\mathrm{\Delta }\tilde{r}}{2{\tilde{r}}_J}\left]\right[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
midpoint
	$=$	$x_J\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]+\{x_J-x_{J-1}\left\}\right[-(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3-\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}-\frac{5}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{3}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4+\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^5]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-\frac{1}{2}+\frac{3}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2+\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[\frac{1}{4}+\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)+\frac{3}{2}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]$
		$+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[-\frac{1}{8}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}\right)-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^2-\frac{3}{4}(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^3-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4-\frac{1}{8}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^5]$
---- next in line ----
	$=$	$x_J[-2(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]+x_{J-1}[(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]+(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[2(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2-\frac{1}{2}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
	$=$	$x_J[-2(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]+\{2x_{J-1}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3-2x_{J-1}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3\}+x_{J-1}[(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2+(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3+\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right]+2(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}[(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2-\frac{1}{4}(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4]$
$\Rightarrow {𝒜}\cdot x_{J+1}$	$=$	$(x_J-x_{J-1})[-2(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^3]+{𝒜}\cdot x_{J-1}+2(x_J{)}^{\text{'}}\mathrm{\Delta }\tilde{r}\left[\right(\frac{\mathrm{\Delta }\tilde{r}}{\tilde{r}}{)}^2-\frac{1}{4}\left(\frac{\mathrm{\Delta }\tilde{r}}{{\tilde{r}}_J}{)}^4\right].$

Interface

CORE:   When $J=(i-1)$ (where $i$ means interface), we can obtain the fractional displacements at the interface, $x_i$ and $p_i$, via the expressions,

$x_i$

$=$

$x_{i-2}-2\mathrm{\Delta }\tilde{r}\left\{\frac{1}{\tilde{r}}\right[3x+\frac{p}{{\gamma }_g}]{\}}_{i-1},$

and

$p_i$

$=$

$p_{i-2}+2\mathrm{\Delta }\tilde{r}\{\frac{\tilde{\rho }}{\tilde{P}}\cdot \frac{{\tilde{M}}_r}{{\tilde{r}}^2}[(4x+p)+{\sigma }_c^2(\frac{2\pi }{3}\cdot \frac{{\tilde{\rho }}_c{\tilde{r}}^3}{{\tilde{M}}_r})x]{\}}_{i-1}.$

Then, setting $J=i$, the pair of radial derivatives at the interface and as viewed from the perspective of the core is given by the expressions,

$(\frac{dx}{d\tilde{r}}{)}_i{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$-\frac{1}{{\tilde{r}}_i}[3x_i+\frac{p_i}{6/5}],$

and

$(\frac{dp}{d\tilde{r}}{)}_i{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

$=$

$\frac{({\tilde{\rho }}_i{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\tilde{P}}_i}\cdot \frac{{\tilde{M}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\tilde{r}}_i^2}[(4x_i+p_i)+{\sigma }_c^2(\frac{2\pi }{3}\cdot \frac{{\tilde{\rho }}_c{\tilde{r}}_i^3}{{\tilde{M}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}\left)x_i\right].$

It is important to recognize that, throughout the core, $(dx/d\tilde{r})$ has been evaluated by setting ${\gamma }_g=6/5$. If we continue to use this value of ${\gamma }_g$ at the interface, we are determining the slope as viewed from the perspective of the core.

ENVELOPE:   On the other hand, as viewed from the perspective of the envelope, all parameters used to determine $(dx/d\tilde{r}{)}_i$ at the interface (and throughout the entire envelope) are the same except ${\gamma }_g$, which equals 2 instead of 6/5. Specifically at the interface, we have,

$(\frac{dx}{d\tilde{r}}{)}_i{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$-\frac{1}{{\tilde{r}}_i}[3x_i+\frac{p_i}{2}],$

and

$(\frac{dp}{d\tilde{r}}{)}_i{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$

$=$

$\frac{({\tilde{\rho }}_i{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}}{{\tilde{P}}_i}\cdot \frac{{\tilde{M}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\tilde{r}}_i^2}[(4x_i+p_i)+{\sigma }_c^2(\frac{2\pi }{3}\cdot \frac{{\tilde{\rho }}_c{\tilde{r}}_i^3}{{\tilde{M}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}\left)x_i\right].$

(See, for example, our related discussion.) Hence, we appreciate that there is a discontinuous change in the value of this slope at the interface. We note as well — for the first time (8/17/2023)! — that there must also be a discontinuous jump in the slope of the "pressure perturbation." All of the variables used to evaluate $(dp/d\tilde{r}{)}_i$ are the same irrespective of your core/envelope point of view except the leading density term. As viewed from the perspective of the core, $({\tilde{\rho }}_i){|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=m_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5({\mu }_e/{\mu }_c{)}^{-10}{\theta }_i^5$ whereas, from the perspective of the envelope, $({\tilde{\rho }}_i){|}_{\mathrm{e}\mathrm{n}\mathrm{v}}=m_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^5({\mu }_e/{\mu }_c{)}^{-9}{\theta }_i^5{\varphi }_i$. Appreciating that ${\varphi }_i=1$, this means that the slope of the "pressure perturbation" is a factor of ${\mu }_e/{\mu }_c$ smaller as viewed from the perspective of the envelope.

Then the value of the fractional radial displacement and the value of the pressure perturbation at the first zone outside of the interface are obtained by setting $J=i$. That is,

$x_{i+1}$

$=$

$x_{i-1}-2\mathrm{\Delta }\tilde{r}\left\{\frac{1}{\tilde{r}}\right[3x+\frac{p}{2}]{\}}_i,$

and

$p_{i+1}$

$=$

$p_{i-1}+2\mathrm{\Delta }\tilde{r}\{\frac{({\tilde{\rho }}_i){|}_{\mathrm{e}\mathrm{n}\mathrm{v}}}{\tilde{P}}\cdot \frac{{\tilde{M}}_r}{{\tilde{r}}^2}[(4x+p)+{\sigma }_c^2(\frac{2\pi }{3}\cdot \frac{{\tilde{\rho }}_c{\tilde{r}}^3}{{\tilde{M}}_r})x]{\}}_i.$

But, as written, these two expressions are unacceptable because the values just inside the interface, $x_{i-1}$ and $p_{i-1}$, are not known as viewed from the perspective of the envelope. However, we can fix this by drawing from the "average" expressions as replacements, namely,

$x_i$

$=$

$\frac{1}{2}(x_{i-1}+x_{i+1})\Rightarrow x_{i-1}=(2x_i-x_{i+1}),$

and

$p_i$

$=$

$\frac{1}{2}(p_{i-1}+p_{i+1})\Rightarrow p_{i-1}=(2p_i-p_{i+1}),$

in which case we have,

$2x_{i+1}$

$=$

$2x_i-2\mathrm{\Delta }\tilde{r}\left\{\frac{1}{\tilde{r}}\right[3x+\frac{p}{2}]{\}}_i,$

and

$2p_{i+1}$

$=$

$2p_i+2\mathrm{\Delta }\tilde{r}\{\frac{({\tilde{\rho }}_i){|}_{\mathrm{e}\mathrm{n}\mathrm{v}}}{\tilde{P}}\cdot \frac{{\tilde{M}}_r}{{\tilde{r}}^2}[(4x+p)+{\sigma }_c^2(\frac{2\pi }{3}\cdot \frac{{\tilde{\rho }}_c{\tilde{r}}^3}{{\tilde{M}}_r})x]{\}}_i.$

See Also