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OdeRungeKutta.f95
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! adapode, Copyright (C) 2013 Michael Reed
MODULE OdeRungeKutta
USE Constants
USE Functions
IMPLICIT NONE
CONTAINS
PURE SUBROUTINE AdaptiveMethod(F,x,xout,h,m,b,c,ce,e)
IMPLICIT NONE
INTEGER::r
INTEGER,INTENT(IN)::m
REAL(dbl),INTENT(IN)::h
REAL(dbl),INTENT(OUT)::e
REAL(dbl),DIMENSION(:),INTENT(IN)::x,c,ce
REAL(dbl),DIMENSION(:,:),INTENT(IN)::b
REAL(dbl),DIMENSION(SIZE(x))::p,s
REAL(dbl),DIMENSION(SIZE(x)),INTENT(OUT)::xout
REAL(dbl),DIMENSION(m,SIZE(x))::K
INTERFACE
PURE FUNCTION F(x)
USE Constants
IMPLICIT NONE
REAL(dbl),INTENT(IN),DIMENSION(:)::x
REAL(dbl),DIMENSION(SIZE(x))::F
END FUNCTION
END INTERFACE
K = ExplicitK(F,x,h,b,m)
p = c(1)*K(1,:)
s = ce(1)*K(1,:)
DO r = 2,m
p = p + c(r)*K(r,:)
s = s + ce(r)*K(r,:)
END DO
e = MAXVAL(DABS(h*s))
xout = x + h*p
END SUBROUTINE AdaptiveMethod
PURE FUNCTION GeneralMethod(F,x,h,m,b,c)
IMPLICIT NONE
INTEGER::r
INTEGER,INTENT(IN)::m
REAL(dbl),INTENT(IN)::h
REAL(dbl),DIMENSION(:),INTENT(IN)::x,c
REAL(dbl),DIMENSION(:,:),INTENT(IN)::b
REAL(dbl),DIMENSION(SIZE(x))::p,GeneralMethod
REAL(dbl),DIMENSION(m,SIZE(x))::K
INTERFACE
PURE FUNCTION F(x)
USE Constants
IMPLICIT NONE
REAL(dbl),INTENT(IN),DIMENSION(:)::x
REAL(dbl),DIMENSION(SIZE(x))::F
END FUNCTION
END INTERFACE
K = ExplicitK(F,x,h,b,m)
p = c(1)*K(1,:)
DO r = 2,m
p = p + c(r)*K(r,:)
END DO
GeneralMethod = x + h*p
END FUNCTION GeneralMethod
PURE FUNCTION ExplicitK(F,x,h,b,m)
USE Functions
IMPLICIT NONE
INTEGER::r,s
INTEGER,INTENT(IN)::m
REAL(dbl),INTENT(IN)::h
REAL(dbl),DIMENSION(:,:),INTENT(IN)::b
REAL(dbl),DIMENSIOn(:),INTENT(IN)::x
REAL(dbl),DIMENSION(SIZE(x))::z
REAL(dbl),DIMENSION(m,SIZE(x))::ExplicitK
INTERFACE
PURE FUNCTION F(x)
USE Constants
IMPLICIT NONE
REAL(dbl),INTENT(IN),DIMENSION(:)::x
REAL(dbl),DIMENSION(SIZE(x))::F
END FUNCTION
END INTERFACE
ExplicitK(1,:) = F(x)
DO r = 2,m
z = 0.d0
DO s = 1,r-1
z = z + b(r,s)*ExplicitK(s,:)
END DO
ExplicitK(r,:) = F(x + h*z)
END DO
RETURN
END FUNCTION ExplicitK
PURE SUBROUTINE Midpoint2nd(m,brs,c)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),brs(:,:)
m = 2
ALLOCATE(c(m))
ALLOCATE(brs(m,m))
c = (/ 0.d0, 1.d0 /)
brs(1,:) = (/ 0.d0 ,0.d0 /)
brs(2,:) = (/ 0.5d0,0.d0 /)
END SUBROUTINE Midpoint2nd
PURE SUBROUTINE Kutta3rd(m,brs,c)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),brs(:,:)
m = 3
ALLOCATE(c(m))
ALLOCATE(brs(m,m))
c = (/ 1.d0, 4.d0, 1.d0/)
c = c / 6.d0
brs(1,:) = (/ 0.d0 , 0.d0, 0.d0 /)
brs(2,:) = (/ 0.5d0, 0.d0, 0.d0 /)
brs(3,:) = (/ -1.d0, 2.d0, 0.d0 /)
END SUBROUTINE Kutta3rd
PURE SUBROUTINE Classical4th(m,brs,c)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),brs(:,:)
m = 4
ALLOCATE(c(m))
ALLOCATE(brs(m,m))
c = (/ 1.d0, 2.d0, 2.d0, 1.d0 /)
c = c / 6.d0
brs(1,:) = (/ 0.d0 ,0.d0 ,0.d0,0.d0 /)
brs(2,:) = (/ 0.5d0,0.d0 ,0.d0,0.d0 /)
brs(3,:) = (/ 0.d0 ,0.5d0,0.d0,0.d0 /)
brs(4,:) = (/ 0.d0 ,0.d0 ,1.d0,0.d0 /)
END SUBROUTINE Classical4th
PURE SUBROUTINE AdaptiveHeunEuler(m,brs,c,ce)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),ce(:),brs(:,:)
m = 2
ALLOCATE(c(m))
ALLOCATE(ce(m))
ALLOCATE(brs(m,m))
ce = (/ 0.5d0 ,0.5d0 /)
c = (/ 1.d0 ,0.d0 /)
ce = c-ce ! convert constants to compute error term
brs(1,:) = (/ 0.d0 ,0.d0 /)
brs(2,:) = (/ 1.d0 ,0.d0 /)
END SUBROUTINE AdaptiveHeunEuler
PURE SUBROUTINE BogackiShampineRK23(m,brs,c,ce)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),ce(:),brs(:,:)
m = 4
ALLOCATE(c(m))
ALLOCATE(ce(m))
ALLOCATE(brs(m,m))
ce = (/ 2.d0/9.d0, 1.d0/3.d0, 4.d0/9.d0, 0.d0 /)
c = (/ 7.d0/24.d0, 1.d0/4.d0, 1.d0/3.d0, 1.d0/8.d0 /)
ce = c-ce ! convert constants to compute error term
brs(1,:) = (/ 0.d0 ,0.d0 ,0.d0 ,0.d0 /)
brs(2,:) = (/ 0.5d0 ,0.d0 ,0.d0 ,0.d0 /)
brs(3,:) = (/ 0.d0 ,0.75d0 ,0.d0 ,0.d0 /)
brs(4,:) = (/ 2.d0/9.d0 ,1.d0/3.d0 ,4.d0/9.d0 ,0.d0/)
END SUBROUTINE BogackiShampineRK23
PURE SUBROUTINE FehlbergRK45(m,brs,c,ce)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),ce(:),brs(:,:)
m = 6
ALLOCATE(c(m))
ALLOCATE(ce(m))
ALLOCATE(brs(m,m))
ce = (/ 25.d0/216.d0, 0.d0, 1408.d0/2565.d0, 2197.d0/4104.d0, -0.2d0, 0.d0 /)
c = (/ 16.d0/135.d0, 0.d0, 6656.d0/12825.d0, 28561.d0/56430.d0, &
-0.18d0, 2.d0/55.d0 /)
ce = c-ce ! convert constants to compute error term
brs(1,:) = (/ 0.d0 ,0.d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(2,:) = (/ 0.25d0 ,0.d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(3,:) = (/ 0.09375d0 ,0.28125d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(4,:) = (/ 1932.d0/2197.d0 ,-7200.d0/2197.d0 ,7296.d0/2197.d0 ,&
0.d0 ,0.d0 ,0.d0 /)
brs(5,:) = (/ 439.d0/216.d0 ,-8.d0 ,3680.d0/513.d0 ,&
-845.d0/4104.d0 ,0.d0 ,0.d0/)
brs(6,:) = (/ -8.d0/27.d0 ,2.d0 ,3544.d0/2565.d0 ,&
1859.d0/4104.d0 ,-11.d0/40.d0 ,0.d0/)
END SUBROUTINE FehlbergRK45
PURE SUBROUTINE CashKarpRK45(m,brs,c,ce)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),ce(:),brs(:,:)
m = 6
ALLOCATE(c(m))
ALLOCATE(ce(m))
ALLOCATE(brs(m,m))
ce = (/ 37.d0/378.d0, 0.d0, 250.d0/621.d0, 125.d0/594.d0, 0d0, &
512.d0/1771.d0 /)
c = (/ 2825.d0/27648.d0, 0.d0, 18575.d0/48384.d0, 13525.d0/55296.d0, &
277.d0/14336.d0, 0.25d0 /)
ce = c-ce ! convert constants to compute error term
brs(1,:) = (/ 0.d0 ,0.d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(2,:) = (/ 0.2d0 ,0.d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(3,:) = (/ 3.d0/40.d0 ,9.d0/40.d0 ,0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(4,:) = (/ 3.d0/40.d0 ,-9.d0/10.d0 ,6.d0/5.d0 ,&
0.d0 ,0.d0 ,0.d0/)
brs(5,:) = (/ -11.d0/54.d0 ,5.d0/2.d0 ,-70.d0/27.d0 ,&
35.d0/27.d0 ,0.d0 ,0.d0/)
brs(6,:) = (/ 1631.d0/55296.d0 ,175.d0/512.d0 ,575.d0/13824.d0 ,&
44275.d0/110592.d0,253.d0/4096.d0 ,0.d0/)
END SUBROUTINE CashKarpRK45
PURE SUBROUTINE DormandPrinceRK45(m,brs,c,ce)
IMPLICIT NONE
INTEGER,INTENT(OUT)::m
REAL(dbl),ALLOCATABLE,INTENT(OUT)::c(:),ce(:),brs(:,:)
m = 7
ALLOCATE(c(m))
ALLOCATE(ce(m))
ALLOCATE(brs(m,m))
ce = (/ 5179.d0/57600.d0, 0.d0, 7571.d0/16695.d0, 393.d0/640.d0, &
-92097.d0/339200.d0, 187.d0/2100.d0, 1.d0/40.d0 /)
c = (/ 35.d0/384.d0, 0.d0, 500.d0/1113.d0, 125.d0/192.d0, &
-2187.d0/6784.d0, 11.d0/84.d0, 0.d0 /)
ce = c-ce ! convert constants to compute error term
brs(1,:) = (/ 0.d0 ,0.d0 , 0.d0, 0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(2,:) = (/ 0.2d0 ,0.d0 , 0.d0, 0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(3,:) = (/ 3.d0/40.d0 ,9.d0/40.d0, 0.d0, 0.d0 ,0.d0 ,0.d0, 0.d0 /)
brs(4,:) = (/ 44.d0/45.d0 ,-56.d0/15.d0 ,32.d0/9.d0 ,&
0.d0 ,0.d0 ,0.d0 ,0.d0/)
brs(5,:) = (/ 19372.d0/6561.d0,-25360.d0/2187.d0,64448.d0/6561.d0 ,&
-212.d0/729.d0 ,0.d0 ,0.d0 ,0.d0/)
brs(6,:) = (/ 9017.d0/3168.d0 ,-355.d0/33.d0 ,46732.d0/5247.d0 ,&
49.d0/176.d0 ,-5103.d0/18656.d0,0.d0 ,0.d0/)
brs(7,:) = (/ 35.d0/384.d0 ,0.d0 ,500.d0/1113.d0 ,&
125.d0/192.d0 ,-2187.d0/6784.d0 ,11.d0/84.d0,0.d0 /)
END SUBROUTINE DormandPrinceRK45
END MODULE OdeRungeKutta