{fpe_2D_f.PDE  - FPE PDESolns - RGN's advice: FIRSTPARTS }
title 'time-dependent 2D diffusion' {D.Bakewell Fr/26/5/2000}
SELECT
  regrid=off  { use fixed grid for 2D problems }
!  regrid=on  {enable adaptive grid - warning can take too long}
firstparts
COORDINATES
cartesian (x, y)
VARIABLES
p     {solute probability or concentration}
DEFINITIONS
a=11. {rectangle horizontal dimensions}
b=11. {rectangle vertical dimensions}
c=1.   {rectangle centre origin} 
Cdiff=1.
k=2                         {velocity constant }
Qx=-k/x                  {solute velocity in X direction}
Qy=-k/y              {solute velocity in Y direction}
{ -------- steady state p(x,y) from analytical solution -----}
al2=1.-k/Cdiff
ps= (al2/ (b^al2-c^al2) ) * (al2/ (a^al2-c^al2) ) * ( (x*y)^(al2-1.) )
Fluy1= Cdiff*dy(p)-p*Qy    {for monitoring Flux in Y direction}
Fluxl= Cdiff*dx(p)-p*Qx   {for monitoring Flux in X direction}
Fluy2= Cdiff*dy(p)-p*Qy   {for monitoring Flux in Y direction}
Flux2= Cdiff*dx(p)-p*Qx   {for monitoring Flux in X direction}
J= vector(p*Qy-Cdiff*dy(p),p*Qx-Cdiff*dx(p))  
INITIAL VALUES
p=1/(c-b)*1/(c-a) 
EQUATIONS
! Cdiff*(dxx(p)+dyy(p))-p*dx(Qx)~p*dy(Qy)-Qx*<:Lx(p)~Qy*dy(p) "dttpJ {2DFPE}
Cdiff*div[grad(p)]-dx[p*Qx]-dy[Qy*p]=dt(p)      { 2D FPE equation }
BOUNDARIES
Region 1
natural(p) = 0
start (c,c)
line to (a,c) to (a,b) to (c,b)    to finish         {walk Left outer boundary }
TIME 0 to 400000 by 0.001     { establish time range and initial timestep }
MONITORS
for cycle=10     contour (p)
elevation (p) from (c,c) to (a,c)  as "X-Axis p"
elevation (p) from (c,c) to (c,b)  as "Y-axis p"
elevation(p) from (c,c) to (a,b)  as "XY-x-section p"
elevation(Fluy1) from (c,c) to (a,c) as "X axis: Flux in Y direction"
elevation(Fluy2) from (c,b) to (a,b) as "// X axis @ Y=b Flux in Y dir"
elevation (Flux2) from (c,c) to (c,b) as "Y axis: Flux in X direction"
elevation (Fluxl) from (a,c) to (a,b) as "Y axis @ X=b: Flux in X dir"
contour (Fluy1) contour (Fluxl) contour (Fluy2) contour (Flux2)
contour (ps ) contour (p-ps )
HISTORIES
history (p) at (c,c)
END  26300