I am trying to write a MATLAB code that will apply an expansion factor $($R$)$ to the nodes in the given parameters. Right now, I have the code in terms of a constant expansion $($dx$).$ I would like each iteration to be expanding by a Ratio R$.$

Current code:

clear all; close all; clc;

p$=$mfilename;

$\%1$D steady state convection diffusion hybrid method

$\%$ finite volume

$\%$ uniform mesh

$\%|$Pe central difference

$\%|$Pe$|>2$ upwind

L$=1$;

N$=20$;

dx$=$L$/$N;

gamma$=0\mathrm{.}1$;

rho$=1$;

u$=2\mathrm{.}0$;

Pe$=$rho$*$u$*$L$/$gamma;

phi$\_0=0$;

phi$\_$L$=1$;

F$=$rho$*$u;

D$=$gamma$/$dx;

Pe$\_$x$=$F$/$D;

$\%\%$ Graduates

$\%$coefficients

if Pe$\_$x$>2$

sprintf$(\text{'}$Pe$\_$x $=\%$g$,\%$s$\text{'},$ Pe$\_$x$,$ 'upwind'$)$

A$\_$w $=$ ones$($N$,1)*$F;

A$\_$e $=$ zeros$($N$,1)$;

elseif Pe$\_$x

sprintf$(\text{'}$Pe$\_$x $=\%$g$,\%$s$\text{'},$ Pe$\_$x$,$ 'upwind'$)$

A$\_$w $=$ zeros$($N$,1)$;

A$\_$e $=$ ones$($N$,1)*(-$F$)$;

else

sprintf$(\text{'}$Pe$\_$x $=\%$g$,\%$s$\text{'},$ Pe$\_$x$,$ 'central'$)$

A$\_$w $=$ ones$($N$,1)*($D$+$F$/2)$;

A$\_$e $=$ ones$($N$,1)*($D$-$F$/2)$;

end

q$\_$p $=$ zeros$($N$,1)$;

q$\_$u $=$ zeros$($N$,1)$;

$\%\%$ Undergrad

$\%$ west

if Pe$\_$x$>2$

A$\_$e$(1)=0$;

A$\_$w$(1)=0$;

q$\_$p$(1)=-(2*$D$+$F$)$;

q$\_$u$(1)=(2*$D$+$F$)*$phi$\_0$;

elseif Pe$\_$x

A$\_$e$(1)=-$F;

A$\_$w$(1)=0$;

q$\_$p$(1)=-(2*$D$)$;

q$\_$u$(1)=(2*$D$)*$phi$\_0$;

else

A$\_$e$(1)=($D$-$F$/2)$;

A$\_$w$(1)=0$;

q$\_$p$(1)=-(2*$D$+$F$)$;

q$\_$u$(1)=(2*$D$+$F$)*$phi$\_0$;

end

$\%$east

if Pe$\_$x$>2$

A$\_$e$($N$)=0$;

A$\_$w$($N$)=$F;

q$\_$p$($N$)=-(2*$D$)$;

q$\_$u$($N$)=(2*$D$)*$phi$\_$L;

elseif Pe$\_$x

A$\_$e$($N$)=0$;

A$\_$w$($N$)=0$;

q$\_$p$($N$)=$F$-(2*$D$)$;

q$\_$u$($N$)=-($F$-2*$D$)*$phi$\_$L;

else

A$\_$e$($N$)=0$;

A$\_$w$($N$)=$D $+$ F$/2$;

q$\_$p$($N$)=-2*$D $+$F;

q$\_$u$($N$)=(2*$D$-$ F$)*$phi$\_$L;

end

$\%$ assembly

A$\_$p $=$ A$\_$w $+$ A$\_$e $-$ q$\_$p;

$\%$ TDMA

phi$=$tdma$(-$A$\_$w$,$A$\_$p$,-$A$\_$e$,$q$\_$u$)$;

$\%$ COORDINATES OF CELL CENTERS

x$=$linspace$(0,$L$,$N$+1)$;

for i$=1$:N

xc$($i$)=0\mathrm{.}5*($x$($i$+1)+$x$($i$))$;

end

$\%$ CALCULATE EXACT SOLUTION AND ERROR NORM

ERROR$=0$;

for i$=1$:N

phi$\_$ex$($i$)=$phi$\_0+($exp$($xc$($i$)*$Pe$/$L$)-1)/($exp$($Pe$)-1)*($phi$\_$L$-$phi$\_0)$;

ERROR$=$ERROR$+$abs$($phi$\_$ex$($i$)-$phi$($i$))$;

end

ERROR$=$ERROR$/$N

X$=[$x$(1),$xc$,$x$($end$)]$;

PHI$=[$phi$\_0,$phi',phi$\_$L$]$;

PHI$\_$EX$=[$phi$\_0,$phi$\_$ex$,$phi$\_$L$]$;

plot$($X$,$PHI,$\text{'}$bs$-\text{'},$X$,$PHI$\_$EX$,\text{'}$r$*-\text{'},$'linewidth',$1)$

legend$(\text{'}$FV$\text{'},$'exact'$)$

TDMA Code:

function X $=$ tdma$($A$,$B$,$C$,$D$)$

Cp $=$ C;

Dp $=$ D;

n $=$ length$($A$)$;

X $=$ zeros$($n$,1)$;

Cp$(1)=$ C$(1)/$B$(1)$;

Dp$(1)=$ D$(1)/$B$(1)$;

for i $=2$:n

Cp$($i$)=$ C$($i$)/($B$($i$)-$Cp$($i$-1)*$A$($i$))$;

Dp$($i$)=($D$($i$)-$Dp$($i$-1)*$A$($i$))/($B$($i$)-$Cp$($i$-1)*$A$($i$))$;

end

X$($n$)=$ Dp$($n$)$;

for i $=$ n$-1$:$-1$:$1$

X$($i$)=$ Dp$($i$)-$Cp$($i$)*$X$($i$+1)$;

end:

Graph: $($See attached image$)$

The graph shows evenly distributed iterations along X$-$axis. I need one that expands by a factor of some ratio R$.$