求解問題
Write a program to solve the following ordinary differential equation by
- basic Euler method
- improved Euler method
- four-order Runge-Kutta method
=3%20%5Cend%7Bcases%7D%20x%5Cin%5B0,%201.5%5D)
and calculate
y(1.5)withstepsize=0.1, 0.1/2, 0.1/4, 0.1/8Compare it with analytic solution (in figure)
%20=%20%5Cdfrac%7B3%7D%7B1+x%5E3%7D)
主程序
program main
use ODE
implicit none
real :: x0=0.0,y0=3.0,a=0.0,b=1.5,h=0.1
integer :: n,i,j
character(len=512) :: filename
print *,'The analytic solution of y(1.5)= ',analytic_solution(1.5)
print *
print *,'Basic Euler Method'
print '(5x,a,f3.1,3x,a,f3.1,3x,a,f3.1,3x,a,f3.1)','Set initial x0= ',x0,'y0= ',y0,'a= ',a,'b= ',b
do j=0,3
h=0.1/2**j
print '(5x,a,f6.4)','Set h= ',h
call init(x0,y0,a,b,h)
n=size(x)-1
call Basic_Euler
print '(8x,a,f10.8)','y(1.5)= ',y(n)
write(filename, *) j
filename='BEM_'//Trim(AdjustL(filename))//'.txt'
open(101,file=filename)
write(101,'(f3.1,f10.8)') (x(i),y(i),i=0,n)
close(101)
deallocate(x,y)
end do
print *
print *,'Improved Euler Method'
print '(5x,a,f3.1,3x,a,f3.1,3x,a,f3.1,3x,a,f3.1)','Set initial x0= ',x0,'y0= ',y0,'a= ',a,'b= ',b
do j=0,3
h=0.1/2**j
print '(5x,a,f6.4)','Set h= ',h
call init(x0,y0,a,b,h)
n=size(x)-1
call Improved_Euler
print '(8x,a,f10.8)','y(1.5)= ',y(n)
write(filename, *) j
filename='IEM_'//Trim(AdjustL(filename))//'.txt'
open(101,file=filename)
write(101,'(f3.1,f10.8)') (x(i),y(i),i=0,n)
close(101)
deallocate(x,y)
end do
print *
print *,'Four Order Runge-Kutta Method'
print '(5x,a,f3.1,3x,a,f3.1,3x,a,f3.1,3x,a,f3.1)','Set initial x0= ',x0,'y0= ',y0,'a= ',a,'b= ',b
do j=0,3
h=0.1/2**j
print '(5x,a,f6.4)','Set h= ',h
call init(x0,y0,a,b,h)
n=size(x)-1
call Four_Order_Runge_Kutta
print '(8x,a,f10.8)','y(1.5)= ',y(n)
write(filename, *) j
filename='RKM_'//Trim(AdjustL(filename))//'.txt'
open(101,file=filename)
write(101,'(f3.1,f10.8)') (x(i),y(i),i=0,n)
close(101)
deallocate(x,y)
end do
print *
end program main
求解ODE方程的關鍵方法寫在ODE模塊中：
module ODE
implicit none
private
public :: x,y,init,Basic_Euler,Improved_Euler,Four_Order_Runge_Kutta,analytic_solution
real,allocatable :: x(:),y(:)
real :: x0,y0,a,b,h
integer :: n,i
contains
function f(x,y)
implicit none
real :: f,x,y
f=-x*x*y*y
end function f
function analytic_solution(x) result(f)
implicit none
real :: f,x
f=3.0/(1+x*x*x)
end function
subroutine init(x0_,y0_,a_,b_,h_)
implicit none
real :: x0_,y0_,a_,b_,h_
x0=x0_
y0=y0_
a=a_
b=b_
h=h_
n=int((b-a)/h)
allocate(x(0:n),y(0:n))
x=(/ (a+i*h,i=0,n) /)
y=0
y(0)=y0
end subroutine init
subroutine Basic_Euler()
implicit none
do i=1,n
y(i)=y(i-1)+h*f(x(i-1),y(i-1))
end do
end subroutine Basic_Euler
subroutine Improved_Euler()
implicit none
real :: y_
do i=1,n
y_=y(i-1)+h*f(x(i-1),y(i-1))
y(i)=y(i-1)+h/2*(f(x(i-1),y(i-1))+f(x(i),y_))
end do
end subroutine Improved_Euler
subroutine Four_Order_Runge_Kutta()
implicit none
real :: k1,k2,k3,k4
do i=1,n
k1=f(x(i-1),y(i-1))
k2=f(x(i-1)+h/2,y(i-1)+h/2*k1)
k3=f(x(i-1)+h/2,y(i-1)+h/2*k2)
k4=f(x(i-1)+h,y(i-1)+h*k3)
y(i)=y(i-1)+h/6*(k1+2*k2+2*k3+k4)
end do
end subroutine Four_Order_Runge_Kutta
end module ODE
其中init方法是用來初始化：
subroutine init(x0_,y0_,a_,b_,h_)
implicit none
real :: x0_,y0_,a_,b_,h_
x0=x0_
y0=y0_
a=a_
b=b_
h=h_
n=int((b-a)/h)
allocate(x(0:n),y(0:n))
x=(/ (a+i*h,i=0,n) /)
y=0
y(0)=y0
end subroutine init
analytic_solution是數值解：
function analytic_solution(x) result(f)
implicit none
real :: f,x
f=3.0/(1+x*x*x)
end function
Basic Euler Method
流程圖：
原理：
![][3]
[3]: http://latex.codecogs.com/gif.latex?\begin{cases}%20y(x_{n+1})=y(x_n)+hf(x_n,y(x_n))+O(h^2)%20\%20y(x_0)=y_0%20\end{cases}
代碼：
subroutine Basic_Euler()
implicit none
do i=1,n
y(i)=y(i-1)+h*f(x(i-1),y(i-1))
end do
end subroutine Basic_Euler
輸出結果：
Basic Euler 方法
圖例：
不同h每步迭代結果
Improved Euler Method
流程圖：
原理：
![][4]
[4]: http://latex.codecogs.com/gif.latex?\begin{cases}%20\bar%20y_{n+1}=y_n+hf(x_n,y_n)%20\%20y(x_{n+1})=y(x_n)+\dfrac{h}{2}[f(x_n,y(x_n))+f(x_{n+1},\bar%20y_{n+1})]+O(h^2)%20\%20y(x_0)=y_0%20\end{cases}
代碼：
subroutine Improved_Euler()
implicit none
real :: y_
do i=1,n
y_=y(i-1)+h*f(x(i-1),y(i-1))
y(i)=y(i-1)+h/2*(f(x(i-1),y(i-1))+f(x(i),y_))
end do
end subroutine Improved_Euler
輸出結果：
Improved Euler 方法
圖例：
不同h每步迭代結果
Four Order Runge-Kutta Method
流程圖：
原理：
![][5]
[5]: http://latex.codecogs.com/gif.latex?\begin{cases}%20y_{n+1}=y_n+\dfrac{h}{6}(K_1+2K_2+2K_3+K_4)%20\%20K_1=f(x_n,y_n)%20\%20K_2=f(x_n+\dfrac{h}{2},y_n+\dfrac{h}{2}K_1)%20\%20K_3=f(x_n+\dfrac{h}{2},y_n+\dfrac{h}{2}K_2)%20\%20K_4=f(x_n+h,y_n+hK_3)%20\end{cases}
代碼：
subroutine Four_Order_Runge_Kutta()
implicit none
real :: k1,k2,k3,k4
do i=1,n
k1=f(x(i-1),y(i-1))
k2=f(x(i-1)+h/2,y(i-1)+h/2*k1)
k3=f(x(i-1)+h/2,y(i-1)+h/2*k2)
k4=f(x(i-1)+h,y(i-1)+h*k3)
y(i)=y(i-1)+h/6*(k1+2*k2+2*k3+k4)
end do
end subroutine Four_Order_Runge_Kutta
輸出結果：
4 Order Runge-Kutta 方法
圖例：
不同h每步迭代結果
三種方法結果對比
h=0.1時
h=0.1/8時
可以看出，當h較大時，三種方法的差別還是很大的，當h逐漸減小時，三種方法的結果已基本相同。
