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pde2d.f90
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!>
!> @file pde2d.f90
!>
!> @brief
!>
!> @copyright
!> Copyright (©) 2021 EPFL (Ecole Polytechnique Fédérale de Lausanne)
!> SPC (Swiss Plasma Center)
!>
!> spclibs is free software: you can redistribute it and/or modify it under
!> the terms of the GNU Lesser General Public License as published by the Free
!> Software Foundation, either version 3 of the License, or (at your option)
!> any later version.
!>
!> spclibs is distributed in the hope that it will be useful, but WITHOUT ANY
!> WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
!> FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
!>
!> You should have received a copy of the GNU Lesser General Public License
!> along with this program. If not, see <https://www.gnu.org/licenses/>.
!>
!> @authors
!> (in alphabetical order)
!> @author Trach-Minh Tran <trach-minh.tran@epfl.ch>
!>
PROGRAM main
!
! Solving the following 2d PDE using splines:
!
! -d/dx[x d/dx]f - 1/x[d/dy]^2 f = 4(m+1)x^(m+1)cos(my), with f(x=1,y) = 0
! exact solution: f(x,y) = (1-x^2) x^m cos(my)
!
USE bsplines
USE matrix
USE futils
USE conmat_mod
!
IMPLICIT NONE
INTEGER :: nx, ny, nidbas(2), ngauss(2), mbess, nterms
LOGICAL :: nlppform, nlconmat
INTEGER :: i, j, ij, dimx, dimy, nrank, kl, ku, jder(2), it
DOUBLE PRECISION :: pi, coefx(5), coefy(5)
DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: xgrid, ygrid, rhs, sol
TYPE(spline2d) :: splxy
TYPE(gbmat) :: mat
!
CHARACTER(len=128) :: file='pde2d.h5'
INTEGER :: fid
DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: arr
DOUBLE PRECISION, DIMENSION(:,:), ALLOCATABLE :: bcoef, solcal, solana, errsol
DOUBLE PRECISION :: seconds, mem, dopla
DOUBLE PRECISION :: t0, tmat, tfact, tsolv, tgrid, gflops1, gflops2
INTEGER :: nits=500
!
INTERFACE
SUBROUTINE dismat(spl, mat)
USE bsplines
USE matrix
TYPE(spline2d), INTENT(in) :: spl
TYPE(gbmat), INTENT(inout) :: mat
END SUBROUTINE dismat
SUBROUTINE disrhs(mbess, spl, rhs)
USE bsplines
INTEGER, INTENT(in) :: mbess
TYPE(spline2d), INTENT(in) :: spl
DOUBLE PRECISION, INTENT(out) :: rhs(:)
END SUBROUTINE disrhs
SUBROUTINE meshdist(coefs, x, nx)
DOUBLE PRECISION, INTENT(in) :: coefs(5)
INTEGER, INTENT(iN) :: nx
DOUBLE PRECISION, INTENT(inout) :: x(0:nx)
END SUBROUTINE meshdist
SUBROUTINE ibcmat(mat, ny)
USE matrix
TYPE(gbmat), INTENT(inout) :: mat
INTEGER, INTENT(in) :: ny
END SUBROUTINE ibcmat
SUBROUTINE ibcrhs(rhs, ny)
DOUBLE PRECISION, INTENT(inout) :: rhs(:)
INTEGER, INTENT(in) :: ny
END SUBROUTINE ibcrhs
!!$ SUBROUTINE coefeq_poisson(x, y, idt, idw, c)
!!$ DOUBLE PRECISION, INTENT(in) :: x, y
!!$ INTEGER, INTENT(out) :: idt(:,:), idw(:,:)
!!$ DOUBLE PRECISION, INTENT(out) :: c(:)
!!$ END SUBROUTINE coefeq_poisson
END INTERFACE
!
NAMELIST /newrun/ nx, ny, nidbas, ngauss, mbess, nlppform, nlconmat, &
& coefx, coefy
!===========================================================================
! 1.0 Prologue
!
! Read in data specific to run
!
nx = 8 ! Number of intervals in x
ny = 8 ! Number of intervals in y
nidbas = (/3,3/) ! Degree of splines
ngauss = (/4,4/) ! Number of Gauss points/interval
mbess = 2 ! Exponent of differential problem
nterms = 2 ! Number of terms in weak form
nlppform = .TRUE. ! Use PPFORM for gridval or not
nlconmat = .TRUE. ! Use CONMAT instead of DISMAT
coefx(1:5) = (/1.0d0, 0.d0, 0.d0, 0.d0, 1.d0/) ! Mesh point distribution function
coefy(1:5) = (/1.0d0, 0.d0, 0.d0, 0.d0, 1.d0/) ! Mesh point distribution function
!
READ(*,newrun)
WRITE(*,newrun)
!
! Define grid on x (=radial) & y (=poloidal) axis
!
pi = 4.0d0*ATAN(1.0d0)
ALLOCATE(xgrid(0:nx), ygrid(0:ny))
xgrid(0) = 0.0d0; xgrid(nx) = 1.0d0
CALL meshdist(coefx, xgrid, nx)
ygrid(0) = 0.0d0; ygrid(ny) = 2.d0*pi
CALL meshdist(coefy, ygrid, ny)
!
WRITE(*,'(a/(10f8.3))') 'XGRID', xgrid(0:nx)
WRITE(*,'(a/(10f8.3))') 'YGRID', ygrid(0:ny)
!
! Create hdf5 file
!
CALL creatf(file, fid, 'PDE2D Result File', real_prec='d')
CALL attach(fid, '/', 'NX', nx)
CALL attach(fid, '/', 'NY', ny)
CALL attach(fid, '/', 'NIDBAS1', nidbas(1))
CALL attach(fid, '/', 'NIDBAS2', nidbas(2))
CALL attach(fid, '/', 'NGAUSS1', ngauss(1))
CALL attach(fid, '/', 'NGAUSS2', ngauss(2))
CALL attach(fid, '/', 'MBESS', mbess)
!===========================================================================
! 2.0 Discretize the PDE
!
! Set up spline
!
CALL set_spline(nidbas, ngauss, &
& xgrid, ygrid, splxy, (/.FALSE., .TRUE./), nlppform=nlppform)
!!$ WRITE(*,'(/a,/(12(f8.3)))') 'KNOTS in X', splxy%sp1%knots
!!$ WRITE(*,'(/a,/(12(f8.3)))') 'KNOTS in Y', splxy%sp2%knots
!
! FE matrix assembly
!
nrank = (nx+nidbas(1))*ny ! Rank of the FE matrix
kl = (nidbas(1)+1)*ny -1 ! Number of sub-diagnonals
ku = kl ! Number of super-diagnonals
WRITE(*,'(a,5i6)') 'nrank, kl, ku', nrank, kl, ku
!
CALL init(kl, ku, nrank, nterms, mat)
t0 = seconds()
IF(nlconmat) THEN
CALL conmat(splxy, mat, coefeq_poisson)
ELSE
CALL dismat(splxy, mat)
END IF
tmat = seconds() - t0
CALL putmat(fid, '/MAT0', mat, 'Assembled GB matrice')
ALLOCATE(arr(nrank))
!
! BC on Matrix
!
IF(nrank.LT.100) &
& WRITE(*,'(a/(10(1pe12.3)))') 'Diag. of matrix before BC', mat%val(kl+ku+1,:)
CALL ibcmat(mat, ny)
IF(nrank.LT.100) &
& WRITE(*,'(a/(10(1pe12.3)))') 'Diag. of matrix after BC', mat%val(kl+ku+1,:)
!
! RHS assembly
!
ALLOCATE(rhs(nrank), sol(nrank))
CALL disrhs(mbess, splxy, rhs)
!
! BC on RHS
!
CALL ibcrhs(rhs, ny)
CALL putarr(fid, '/RHS', rhs, 'RHS of linear system')
CALL putmat(fid, '/MAT1', mat, 'GB matrice with BC')
WRITE(*,'(a,1pe12.3)') 'Mem used so far (MB)', mem()
!===========================================================================
! 3.0 Solve the dicretized system
!
t0 = seconds()
CALL factor(mat)
tfact = seconds() - t0
gflops1 = dopla('DGBTRF',nrank,nrank,kl,ku,0) / tfact / 1.d9
t0 = seconds()
CALL bsolve(mat, rhs, sol)
!
! Backtransform of solution
!
sol(1:ny-1) = sol(ny)
!
! Spline coefficients, taking into account of periodicity in y
! Note: in SOL, y was numbered first.
!
dimx = splxy%sp1%dim
dimy = splxy%sp2%dim
ALLOCATE(bcoef(0:dimx-1, 0:dimy-1))
DO j=0,dimy-1
DO i=0,dimx-1
ij = MODULO(j,ny) + i*ny + 1
bcoef(i,j) = sol(ij)
END DO
END DO
WRITE(*,'(a,2(1pe12.3))') ' Integral of sol', fintg(splxy, bcoef)
!
tsolv = seconds() - t0
gflops2 = dopla('DGBTRS',nrank,1,kl,ku,0) / tsolv / 1.d9
CALL putarr(fid, '/SOL', sol, 'Spline coefficients of solution')
!===========================================================================
! 4.0 Check the solution
!
! Check function values computed with various method
!
ALLOCATE(solcal(0:nx,0:ny), solana(0:nx,0:ny), errsol(0:nx,0:ny))
DO i=0,nx
DO j=0,ny
solana(i,j) = (1-xgrid(i)**2) * xgrid(i)**mbess * COS(mbess*ygrid(j))
END DO
END DO
jder = (/0,0/)
!
! Compute PPFORM/BCOEFS at first call to gridval
CALL gridval(splxy, xgrid, ygrid, solcal, jder, bcoef)
!
WRITE(*,'(/a)') '*** Checking solutions'
t0 = seconds()
DO it=1,nits ! nits iterations for timing
CALL gridval(splxy, xgrid, ygrid, solcal, jder)
END DO
tgrid = (seconds() - t0)/REAL(nits)
errsol = solana - solcal
IF( SIZE(bcoef,2) .LE. 10 ) THEN
CALL prnmat('BCOEF', bcoef)
CALL prnmat('SOLANA', solana)
CALL prnmat('SOLCAL', solcal)
CALL prnmat('ERRSOL', errsol)
END IF
WRITE(*,'(a,2(1pe12.3))') 'Relative discretization errors', &
& norm2(errsol) / norm2(solana)
WRITE(*,'(a,1pe12.3)') 'GRIDVAL2 time (s) ', tgrid
!
CALL putarr(fid, '/xgrid', xgrid, 'r')
CALL putarr(fid, '/ygrid', ygrid, '\theta')
CALL putarr(fid, '/sol', solcal, 'Solutions')
CALL putarr(fid, '/solana', solana,'Exact solutions')
CALL putarr(fid, '/errors', errsol, 'Errors')
!
! Check derivatives d/dx and d/dy
!
WRITE(*,'(/a)') '*** Checking gradient'
DO i=0,nx
DO j=0,ny
IF( mbess .EQ. 0 ) THEN
solana(i,j) = -2.0d0 * xgrid(i)
ELSE
solana(i,j) = (-(mbess+2)*xgrid(i)**2 + mbess) * &
& xgrid(i)**(mbess-1) * COS(mbess*ygrid(j))
END IF
END DO
END DO
!
jder = (/1,0/)
CALL gridval(splxy, xgrid, ygrid, solcal, jder)
errsol = solana - solcal
CALL putarr(fid, '/derivx', solcal, 'd/dx of solutions')
WRITE(*,'(a,2(1pe12.3))') 'Error in d/dx', norm2(errsol)
!
DO i=0,nx
DO j=0,ny
solana(i,j) = -mbess * (1-xgrid(i)**2) * xgrid(i)**mbess * SIN(mbess*ygrid(j))
END DO
END DO
!
jder = (/0,1/)
CALL gridval(splxy, xgrid, ygrid, solcal, jder)
CALL putarr(fid, '/derivy', solcal, 'd/dy of solutions')
errsol = solana - solcal
WRITE(*,'(a,2(1pe12.3))') 'Error in d/dy', norm2(errsol)
!===========================================================================
! 9.0 Epilogue
!
WRITE(*,'(/a)') '---'
WRITE(*,'(a,1pe12.3)') 'Matrice construction time (s) ', tmat
WRITE(*,'(a,1pe12.3)') 'Matrice factorisation time (s)', tfact
WRITE(*,'(a,1pe12.3)') 'Backsolve time (s) ', tsolv
WRITE(*,'(a,2f10.3)') 'Factor/solve Gflop/s', gflops1, gflops2
!
DEALLOCATE(xgrid, rhs, sol)
DEALLOCATE(solcal, solana, errsol)
DEALLOCATE(bcoef)
DEALLOCATE(arr)
CALL destroy_sp(splxy)
CALL destroy(mat)
!
CALL closef(fid)
!===========================================================================
!
CONTAINS
SUBROUTINE coefeq_poisson(x, y, idt, idw, c)
DOUBLE PRECISION, INTENT(in) :: x, y
INTEGER, INTENT(out) :: idt(:,:), idw(:,:)
DOUBLE PRECISION, INTENT(out) :: c(:)
!
! Weak form = Int(x*dw/dx*dt/dx + 1/x*dw/dy*dt/dy)dxdy
!
c(1) = x !
idt(1,1) = 1
idt(1,2) = 0
idw(1,1) = 1
idw(1,2) = 0
!
c(2) = 1.d0/x
idt(2,1) = 0
idt(2,2) = 1
idw(2,1) = 0
idw(2,2) = 1
END SUBROUTINE coefeq_poisson
!
!+++
FUNCTION norm2(x)
!
! Compute the 2-norm of array x
!
IMPLICIT NONE
DOUBLE PRECISION :: norm2
DOUBLE PRECISION, INTENT(in) :: x(:,:)
DOUBLE PRECISION :: sum2
INTEGER :: i, j
!
sum2 = 0.0d0
DO i=1,SIZE(x,1)
DO j=1,SIZE(x,2)
sum2 = sum2 + x(i,j)**2
END DO
END DO
norm2 = SQRT(sum2)
END FUNCTION norm2
SUBROUTINE prnmat(label, mat)
CHARACTER(len=*) :: label
DOUBLE PRECISION, DIMENSION(:,:), INTENT(in) :: mat
INTEGER :: i
WRITE(*,'(/a)') TRIM(label)
DO i=1,SIZE(mat,1)
WRITE(*,'(10(1pe12.3))') mat(i,:)
END DO
END SUBROUTINE prnmat
END PROGRAM main
!
!+++
SUBROUTINE meshdist(c, x, nx)
!
! Construct an 1d non-equidistant mesh given a
! mesh distribution function.
!
IMPLICIT NONE
DOUBLE PRECISION, INTENT(in) :: c(5)
INTEGER, INTENT(iN) :: nx
DOUBLE PRECISION, INTENT(inout) :: x(0:nx)
INTEGER :: nintg
DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: xint, fint
DOUBLE PRECISION :: a, b, dx, f0, f1, scal
INTEGER :: i, k
!
a=x(0)
b=x(nx)
nintg = 10*nx
ALLOCATE(xint(0:nintg), fint(0:nintg))
!
! Mesh distribution
!
dx = (b-a)/REAL(nintg)
xint(0) = a
fint(0) = 0.0d0
f1 = fdist(xint(0))
DO i=1,nintg
f0 = f1
xint(i) = xint(i-1) + dx
f1 = fdist(xint(i))
fint(i) = fint(i-1) + 0.5*(f0+f1)
END DO
!
! Normalization
!
scal = REAL(nx) / fint(nintg)
fint(0:nintg) = fint(0:nintg) * scal
!!$ WRITE(*,'(a/(10f8.3))') 'FINT', fint
!
! Obtain mesh point by (inverse) interpolation
!
k = 1
DO i=1,nintg-1
IF( fint(i) .GE. REAL(k) ) THEN
x(k) = xint(i) + (xint(i+1)-xint(i))/(fint(i+1)-fint(i)) * &
& (k-fint(i))
k = k+1
END IF
END DO
!
DEALLOCATE(xint, fint)
CONTAINS
DOUBLE PRECISION FUNCTION fdist(x)
DOUBLE PRECISION, INTENT(in) :: x
fdist = c(1) + c(2)*x + c(3)*EXP(-((x-c(4))/c(5))**2)
END FUNCTION fdist
END SUBROUTINE meshdist
!+++
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