abc.txt  see WEB page for development CMSC 455 supplemental
         The problem must be well posed and the solution must be
         continuous and continuously differentiable.

The PDE to be solved is:
 a1(x,y)*uxx(x,y) + b1(x,y)*uxy(x,y) + c1(x,y)*uyy(x,y) +
 d1(x,y)*ux(x,y) + e1(x,y)*uy(x,y)  + f1(x,y) = c(x,y)

where, for this test case
 a1(x,y) = exp(x/2)*exp(y)/2
 b1(x,y) = 0.7/(x*x*y*y + 0.5)
 c1(x,y) = (4 - exp(x) - exp(y/2))*2
 d1(x,y) = x*x+y
 e1(x,y) = x*y*y
 f1(x,y) = 3.0*x + 2.0*y

The solution (normally only the boundary values are computed)
 u(x,y) = x^3 + 2y^3 + 3x^2 y + 4xy^2 + 5xy + 6x + 7y + 8

where
 uxx(x,y) = 6.0*x + 6.0*y
 uxy(x,y) = 6.0*x + 8.0*y + 5.0
 uyy(x,y) = 12.0*y + 8.0*x
 ux(x,y)  = 3.0*x*x + 6.0*x*y + 4.0*y*y + 5.0*y + 6.0
 uy(x,y)  = 6.0*y*y + 3.0*x*x + 8.0*x*y + 5.0*x + 7.0

Thus compute (I used Maple)
 c(x,y) = 0.5*exp(x/2.0)*exp(y)*(6.0*x+6.0*y)             +
     0.7*(6.0*x + 8.0*y + 5.0)/(x*x*y*y+0.5)              +
     (8.0 - 2.0*exp(x) - 2.0*exp(y/2.0))*(12.0*y + 8.0*x) +
     (x*x+y)*(3.0*x*x + 6.0*x*y + 4.0*y*y + 5.0*y + 6.0)  +
     x*y*y*(6.0*y*y + 3.0*x*x + 8.0*x*y + 5.0*x +7.0)     +
     3.0*x + 2.0*y;

Discretization of the PDE for nd=5, gives the terms
 a1(x,y)*uxx(x,y) = a1(x,y) * (1/(12*hx*hx)) *
 (-1*u(x-2hx,y) +16*u(x-hx,y) -30*u(x,y) +16*u(x+hx,y) -1*u(x+2hx,y))

 b1(x,y)*uxy(x,y) = b1(xmin+i*hx,y+j*hy) * (1/(144*hy*hy)) *
 ( 1*u(x-2hx,y-2hy)  -8*u(x-hx,y-2hy) +0*u(x,y-2hy)  +8*u(x+hx,y-2hy)
  -1*u(x+2hx,y-2hy)
  -8*u(x-2hx,y-hy ) +64*u(x-hx,y-hy ) +0*u(x,y-hy ) -64*u(x+hx,y-hy )
  +8*u(x+2hx,y-hy )
  +0*u(x-2hx,y    )  +0*u(x-hx,y    ) +0*u(x,y    )  +0*u(x+hx,y    )
  +0*u(x+2hx,y    )
  +8*u(x-2hx,y+hy ) -64*u(x-hx,y+hy ) +0*u(x,y+hy ) +64*u(x+hx,y+hy )
  -8*u(x+2hx,y+hy )
  -1*u(x-2hx,y+2hy)  +8*u(x-hx,y+2hy) +0*u(x,y+2hy)  -8*u(x+hx,y+2hy)
  +1*u(x+2hx,y+2hy))

 c1(x,y)*uyy(x,y) = c1(x,y) * (1/(12*hy*hy)) *
 (-1*u(x,y-2hy) +16*u(x,y-hy) -30*u(x,y) +16*u(x,y+hy) -1*u(x,y+2hy))

 d1(x,y)*ux(x,y)  =  d1(xmin+i*hx,ymin+j*hy) * (1/(12*hx) *
  (1*u(x-2hx,y) -8*u(x-hx,y) +0*u(x,y) +8*u(x+hx,y) -1*u(x+2hx,y))

 e1(x,y)*uy(x,y)  =  e1(xmin+i*hx,ymin+j*hy) * (1/(12*hy) *
  (1*u(x,y-2hy) -8*u(x,y-hy) +0*u(x,y) +8*u(x,y+hy) -1*u(x,y+2hy))

 discretization for the general central points with indices is
 a1(xmin+i*hx,ymin+j*hy) * (1/(12*hx*hy)) *
  (-1*u(i-2,j) +16*u(i-1,j) -30*u(i,j) +16*u(i+1,j) -1*u(i+2,j))          +
 b1(xmin+i*hx,y+j*hy) * (1/(144*hy*hy)) *
  ( 1*u(i-2,j-2)  -8*u(i-1,j-2) +0*u(i,j-2)  +8*u(i+1,j-2) -1*u(i+2,j-2)
   -8*u(i-2,j-1) +64*u(i-1,j-1) +0*u(i,j-1) -64*u(i+1,j-1) +8*u(i+2,j-1)
   +0*u(i-2,j  )  +0*u(i-1,j  ) +0*u(i,j  )  +0*u(i+1,j  ) +0*u(i+2,j  )
   +8*u(i-2,j+1) -64*u(i-1,j+1) +0*u(i,j+1) +64*u(i+1,j+1) -8*u(i+2,j+1)
   -1*u(i-2,j+2)  +8*u(i-1,j+2) +0*u(i,j+2)  -8*u(i+1,j+2) +1*u(i+2,j+2)) +
 c1(xmin+i*hx,ymin+j*hy) * (1/(12*hy*hy)) *
  (-1*u(i,j-2) +16*u(i,j-1) -30*u(i,j) +16*u(i,j+1) -1*u(i,j+2))          +
 d1(xmin+i*hx,ymin+j*hy) * (1/(12*hx) *
  (1*u(i-2,j) -8*u(i-1,j) +0*u(i,j) +8*u(i+1,j) -1*u(i+2,j))              +
 e1(xmin+i*hx,ymin+j*hy) * (1/(12*hy) *
  (1*u(i,j-2) -8*u(i,j-1) +0*u(i,j) +8*u(i,j+1) -1*u(i,j+2))              +
 f1(xmin+i*hx,ymin+j*hy)                                                  =
 c(xmin+i*hx,ymin+j*hy)

The solver uses  #include "abc.h" to gain access to the users problem

See the file "abc.h" for the required user provided information.