D302 Fast Partial Differential Equation Solver

ELPAHY uses fast Fourier transform techniques for the solution, over a rectangular domain, of the following elliptic, parabolic or hyperbolic part differential equation:

$\{d2\phi (x,y)dx2\}+c$₁{d²φ(x,y)dy²}+c₂{dφ(x,y)dy}+c₃φ(x,y) = ρ(x,y)

where $\phi (x,y)$ is the unknown function, $\rho (x,y)$ the known source term, and $c$₁,c₂,c₃ given coefficients. A large variety of boundary conditions can be specified on the sides of the rectangle.

Structure:

SUBROUTINE subprogram
User Entry Names: ELPAHY
Internal Entry Names: NEWRO, ELANAL, ESOLVE, SYNT, MFT
External References: RFT (D700)
COMMON Block Names and Lengths: /FW1/ 774, /FW2/ 100

Usage:

    CALL ELPAHY(F,NX,NY,DX,DY,C,IBX,BWEST,BEAST,JBY,BSOUTH,BNORTH)

F

( REAL) Two-dimensional array, dimensioned (NX,NY) in the calling program. On input it contains the source term $\rho (x,y)$ and on return it contains the unknown function $\phi (x,y)$ .

NX

( INTEGER) Number of divisions along X. NX must be of the form $2n+1$ .

NY

( INTEGER) Number of divisions along Y.

DX

( REAL) Mesh spacing along X.

DY

( REAL) Mesh spacing along Y.

C

( REAL) One-dimensional array of dimension 3, containing the coefficients $c$₁,c₂,c₃ .

IBX

( INTEGER) Controls the type of boundary conditions on the left (BWEST) and right (BEAST) sides of the rectangular domain:
$IBX=1:$ Imposed periodicity along x; BWEST, BEAST not given.
$IBX=2:$ Given derivative on either vertical side.
$IBX=3:$ Given value on either vertical side.
$IBX=4:$ Given value on the left side, given derivative on the right side.

BWEST

( REAL) One-dimensional array of size NY containing values or derivatives for the left side; the interpretation depends on IBX.

BEAST

( REAL) One-dimensional array of size NY containing values or derivatives for the right side; the interpretation depends on IBX.

JBY

( INTEGER) Controls the type of boundary conditions on the lower (BSOUTH) und upper (BNORTH) sides of the rectangular domain:
Elliptic equation ($c$₁>0 ):
$JBY=1:$ Given value on both lower and upper sides.
$JBY=2:$ Given derivative on both lower and upper sides.
$JBY=3:$ Given value on lower side, given derivative on upper side.
$JBY=4:$ Given derivative on lower side, given value on upper side.
Parabolic equation ($c$₁=0 ):
Specify BSOUTH array only. (If y=time, BSOUTH are initial values and the future BNORTH cannot be specified).
$JBY=1:$ Given value on lower side.
$JBY=2:$ Given derivative on lower side.
Hyperbolic equation ($c$₁<0 ):
The BSOUTH array specifies the value, the BNORTH array the derivative.
$JBY=1$ .

BSOUTH

( REAL) One-dimensional array of size NX containing values or derivatives for the lower side; the interpretation depends on JBY.

BNORTH

( REAL) One-dimensional array of size NX containing values or derivatives for the upper side; the interpretation depends on JBY.

Notes:

If $NX\; >\; 65$ , specify COMMON /FWORK/ of length 6*NX and COMMON /FW1/ of length 6*NX in the calling program. If $NY\; >\; 50$ , specify COMMON /FW2/ of length 2*NY. In either case, make sure your program is loaded before ELPAHY (D302) (this is automatic unless you recompile D302 in the same job).

References:

1. R.C. Le Bail, Use of fast Fourier transforms for solving partial differential equations in physics, J. Comput. Phys. 9 (1972) 440--465

D401

K.S. Kölbig