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C --------------------------------------------------------------------
C SUNDIALS Copyright Start
C Copyright (c) 2002-2022, Lawrence Livermore National Security
C and Southern Methodist University.
C All rights reserved.
C
C See the top-level LICENSE and NOTICE files for details.
C
C SPDX-License-Identifier: BSD-3-Clause
C SUNDIALS Copyright End
C --------------------------------------------------------------------
C FCVODE Example Problem: Advection-diffusion, band linear solver
C with banded user Jacobian.
C
C The following is a simple example problem with a banded
C Jacobian. The problem is the semi-discrete form of the
C advection-diffusion equation in 2D:
C du/dt = d^2 u / dx^2 + .5 du/dx + d^2 u / dy^2
C on the rectangle 0 <= x <= 2, 0 <= y <= 1, and the time
C interval 0 <= t <= 1. Homogeneous Dirichlet boundary conditions
C are posed, and the initial condition is the following:
C u(x,y,t=0) = x(2-x)y(1-y)exp(5xy) .
C The PDE is discretized on a uniform MX+2 by MY+2 grid with
C central differencing, and with boundary values eliminated,
C leaving an ODE system of size NEQ = MX*MY.
C This program solves this problem with CVODE, using the Fortran/C
C interface routine package. This solution uses the BDF method,
C a user-supplied banded Jacobian routine, and scalar relative and
C absolute tolerances. It prints results at t = .1, .2, ..., 1.0.
C At the end of the run, various counters of interest are printed.
C --------------------------------------------------------------------
C
IMPLICIT NONE
C
INTEGER*4 MX, MY, MXMY
PARAMETER (MX=10, MY=5)
PARAMETER (MXMY=MX*MY)
C
DOUBLE PRECISION XMAX, YMAX
DATA XMAX/2.0D0/, YMAX/1.0D0/
C
INTEGER*4 LNST, LNFE, LNSETUP, LNNI, LNCF, LNETF, LNJE
INTEGER*4 IER, METH, IATOL, ITASK, JOUT
C The following declaration specification should match C type long int.
INTEGER*8 NEQ, IOUT(25), IPAR(2), MU, ML
DOUBLE PRECISION RTOL, ATOL, T0, T, TOUT, DTOUT, UNORM
DOUBLE PRECISION U(MXMY), ROUT(10), RPAR(5)
C
DATA LNST/3/, LNFE/4/, LNETF/5/, LNCF/6/, LNNI/7/, LNSETUP/8/,
1 LNJE/17/
C
CALL INITBX(XMAX, YMAX, MX, MY, U, IPAR, RPAR)
C
NEQ = MX*MY
T0 = 0.0D0
METH = 2
IATOL = 1
RTOL = 0.0D0
ATOL = 1.0D-5
MU = MY
ML = MY
DTOUT = 0.1D0
ITASK = 1
C
WRITE(6,10) NEQ
10 FORMAT('Band example problem:'//
1 ' Advection-diffusion, NEQ = ', I2//)
C
CALL FNVINITS(1, NEQ, IER)
IF (IER .NE. 0) THEN
WRITE(6,20) IER
20 FORMAT(///' SUNDIALS_ERROR: FNVINITS returned IER = ', I5)
STOP
ENDIF
C
C initialize banded matrix module
call FSUNBANDMATINIT(1, NEQ, MU, ML, MU+ML, IER)
IF (IER .NE. 0) THEN
WRITE(6,25) IER
25 FORMAT(///' SUNDIALS_ERROR: FSUNBANDEMATINIT IER = ', I5)
STOP
ENDIF
C
C initialize banded linear solver module
call FSUNBANDLINSOLINIT(1, IER)
IF (IER .NE. 0) THEN
WRITE(6,28) IER
28 FORMAT(///' SUNDIALS_ERROR: FSUNBANDLINSOLINIT IER = ', I5)
STOP
ENDIF
C
CALL FCVMALLOC(T0, U, METH, IATOL, RTOL, ATOL,
1 IOUT, ROUT, IPAR, RPAR, IER)
IF (IER .NE. 0) THEN
WRITE(6,30) IER
30 FORMAT(///' SUNDIALS_ERROR: FCVMALLOC returned IER = ', I5)
STOP
ENDIF
C
C attach matrix and linear solver modules to CVLs interface
CALL FCVLSINIT(IER)
IF (IER .NE. 0) THEN
WRITE(6,40) IER
40 FORMAT(///' SUNDIALS_ERROR: FCVLSINIT returned IER = ',I5)
CALL FCVFREE
STOP
ENDIF
C
CALL FCVBANDSETJAC(1, IER)
IF (IER .NE. 0) THEN
WRITE(6,45) IER
45 FORMAT(///' SUNDIALS_ERROR: FCVDENSESETJAC returned IER = ',I5)
CALL FCVFREE
STOP
ENDIF
C
CALL MAXNORM(NEQ, U, UNORM)
WRITE(6,48) T0, UNORM
48 FORMAT(' At t = ', F6.2, ' max.norm(u) = ', E14.6)
C
TOUT = DTOUT
DO 70 JOUT = 1, 10
C
CALL FCVODE(TOUT, T, U, ITASK, IER)
C
CALL MAXNORM(NEQ, U, UNORM)
WRITE(6,50) T, UNORM, IOUT(LNST)
50 FORMAT(' At t = ', F6.2, ' max.norm(u) = ', E14.6,
1 ' NST = ', I4)
C
IF (IER .NE. 0) THEN
WRITE(6,60) IER, IOUT(15)
60 FORMAT(///' SUNDIALS_ERROR: FCVODE returned IER = ', I5, /,
1 ' Linear Solver returned IER = ', I5)
CALL FCVFREE
STOP
ENDIF
C
TOUT = TOUT + DTOUT
70 CONTINUE
C
WRITE(6,80) IOUT(LNST), IOUT(LNFE), IOUT(LNJE), IOUT(LNSETUP),
1 IOUT(LNNI), IOUT(LNCF), IOUT(LNETF)
80 FORMAT(//'Final statistics:'//
1 ' No. steps = ', I4, ' No. f-s = ', I4,
2 ' No. J-s = ', I4, ' No. LU-s = ', I4/
3 ' No. nonlinear iterations = ', I4/
4 ' No. nonlinear convergence failures = ', I4/
5 ' No. error test failures = ', I4)
C
CALL FCVFREE
C
STOP
END
C ----------------------------------------------------------------
SUBROUTINE INITBX(XMAX, YMAX, MX, MY, U0, IPAR, RPAR)
C Load IPAR and RPAR with problem constants and U0 with initial values
IMPLICIT NONE
C
C The following declaration specification should match C type long int.
INTEGER*8 IPAR(*)
INTEGER*4 MX, MY
DOUBLE PRECISION XMAX, YMAX, U0(MY,MX), RPAR(*)
C
INTEGER*4 I, J
DOUBLE PRECISION DX, DY, X, Y, HDCOEF, HACOEF, VDCOEF
C
C Problem constants
DX = XMAX / (MX + 1)
DY = YMAX / (MY + 1)
HDCOEF = 1.0D0 / (DX * DX)
HACOEF = 0.5D0 / (2.0D0 * DX)
VDCOEF = 1.0D0 / (DY * DY)
C Load constants in IPAR and RPAR
IPAR(1) = MX
IPAR(2) = MY
RPAR(1) = DX
RPAR(2) = DY
RPAR(3) = HDCOEF
RPAR(4) = HACOEF
RPAR(5) = VDCOEF
C
C Loop over grid and load initial values.
DO 20 I = 1, MX
X = I * DX
DO 10 J = 1, MY
Y = J * DY
U0(J,I) = X * (XMAX - X) * Y * (YMAX - Y) *
* EXP(5.0D0 * X * Y)
10 CONTINUE
20 CONTINUE
C
RETURN
END
SUBROUTINE MAXNORM(N, U, UNORM)
C Compute max-norm of array U
IMPLICIT NONE
C
INTEGER*8 I, N
DOUBLE PRECISION U(*), UNORM, TEMP
C
TEMP = 0.0D0
DO 10 I = 1, N
TEMP = MAX(ABS(U(I)), TEMP)
10 CONTINUE
UNORM = TEMP
RETURN
END
C ----------------------------------------------------------------
SUBROUTINE FCVFUN(T, U, UDOT, IPAR, RPAR, IER)
C Right-hand side routine
IMPLICIT NONE
C
DOUBLE PRECISION T, U(*), UDOT(*), RPAR(*)
C The following declaration specification should match C type long int.
INTEGER*8 IPAR(*)
INTEGER*4 IER
C
INTEGER*4 I, MX, IOFF, MY, J, IJ
DOUBLE PRECISION UIJ, UDN, UUP, ULT, URT, HDIFF, HADV, VDIFF
DOUBLE PRECISION DX, DY, HDCOEF, HACOEF, VDCOEF
C
C Extract constants from IPAR and RPAR
MX = IPAR(1)
MY = IPAR(2)
DX = RPAR(1)
DY = RPAR(2)
HDCOEF = RPAR(3)
HACOEF = RPAR(4)
VDCOEF = RPAR(5)
C
C Loop over all grid points.
DO 20 I = 1, MX
IOFF = (I - 1) * MY
DO 10 J = 1, MY
C
C Extract u at x_i, y_j and four neighboring points.
IJ = J + IOFF
UIJ = U(IJ)
UDN = 0.0D0
IF (J .NE. 1) UDN = U(IJ - 1)
UUP = 0.0D0
IF (J .NE. MY) UUP = U(IJ + 1)
ULT = 0.0D0
IF (I .NE. 1) ULT = U(IJ - MY)
URT = 0.0D0
IF (I .NE. MX) URT = U(IJ + MY)
C
C Set diffusion and advection terms and load into UDOT.
HDIFF = HDCOEF * (ULT - 2.0D0 * UIJ + URT)
HADV = HACOEF * (URT - ULT)
VDIFF = VDCOEF * (UUP - 2.0D0 * UIJ + UDN)
UDOT(IJ) = HDIFF + HADV + VDIFF
10 CONTINUE
20 CONTINUE
C
IER = 0
C
RETURN
END
C ----------------------------------------------------------------
SUBROUTINE FCVBJAC(N, MU, ML, MDIM, T, U, FU,
1 BJAC, H, IPAR, RPAR, V1, V2, V3, IER)
C Load banded Jacobian
IMPLICIT NONE
C
C The following declaration specification should match C type long int.
INTEGER*8 N, MU, ML, MDIM, IPAR(*)
INTEGER*4 IER
DOUBLE PRECISION T, U(*), FU(*), BJAC(MDIM,*), H, RPAR(*)
DOUBLE PRECISION V1(*), V2(*), V3(*)
C
INTEGER*4 MBAND, MX, MY
INTEGER*4 I, J, K, IOFF, MU1, MU2
DOUBLE PRECISION DX, DY, HDCOEF, HACOEF, VDCOEF
C
C Extract constants from IPAR and RPAR
MX = IPAR(1)
MY = IPAR(2)
DX = RPAR(1)
DY = RPAR(2)
HDCOEF = RPAR(3)
HACOEF = RPAR(4)
VDCOEF = RPAR(5)
C
MU1 = MU + 1
MU2 = MU + 2
MBAND = MU + 1 + ML
C
C Loop over all grid points.
DO 20 I = 1, MX
IOFF = (I - 1) * MY
DO 10 J = 1, MY
K = J + IOFF
C
C Set Jacobian elements in column k of Jb.
BJAC(MU1,K) = -2.0D0 * (VDCOEF + HDCOEF)
IF (I .NE. 1) BJAC(1,K) = HDCOEF + HACOEF
IF (I .NE. MX) BJAC(MBAND,K) = HDCOEF - HACOEF
IF (J .NE. 1) BJAC(MU,K) = VDCOEF
IF (J .NE. MY) BJAC(MU2,K) = VDCOEF
C
10 CONTINUE
20 CONTINUE
C
IER = 0
C
RETURN
END
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