8/7/2019 PC: Euler & Runge - Kutta

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: : 2980: 24.11.2005

PC:

: EulerRunge - Kutta

8/7/2019 PC: Euler & Runge - Kutta

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::

:

xyy =

i.

( ) ey =0 .

ii. EulerRunge-Kutta /. ;

//::

Fortran, Euler Runge-Kutta., compilation , :

Euler,

Euler.out ( )yx, , Gnuplot ( )xfy = .

Runge-Kutta, Runge-Kutta ( )yx, , Gnuplot ( )xfy = .

::

Euler:

( )yxfy ,= nhxx

on +=

( )nnnn yxhfyy ,1 +=+

Runge-Kutta:

( )yxfy ,= nhxx on +=

( )nn yxhfk ,0 =

++=

012

,2

kh

yh

xhfknn

++=

122

,2

kh

yh

xhfknn

( )23 , kyhxhfk nn ++=

( )32101

226

1kkkkyy

nn ++++=+

8/7/2019 PC: Euler & Runge - Kutta

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do i=-n,0write(*,2) x(i), y(i)write(1,2) x(i), y(i)

2 format (2x,F5.2,2X,F8.3)enddo

do i=1,nwrite(*,3) x(i), y(i)write(1,3) x(i), y(i)

3 format (2x,F5.2,2x,F8.3)enddoend

subroutine Runge_Kutta(h,n,x,y)implicit nonereal x(-1000:1000),y(-1000:1000),f,h,k1,k2,k3,k4

integer i,ndo i=0,nk1=f(x(-i),y(-i))k2=f(x(-i)-h/2.,y(-i)+k1*h/2.)k3=f(x(-i)-h/2.,y(-i)+k2*h/2.)k4=f(x(-i)-h,y(-i)+k3*h)y(-i-1)=y(-i)-(k1+2*k2+2*k3+k4)*(h/6.)enddodo i=0,nk1=f(x(i),y(i))k2=f(x(i)+h/2.,y(i)+k1*h/2.)

k3=f(x(i)+h/2.,y(i)+k2*h/2.)k4=f(x(i)+h,y(i)+k3*h)y(i+1)=y(i)+(k1+2*k2+2*k3+k4)*(h/6.)enddoopen (unit=4,file="expdata_Runge-Kutta.out")do i=-n,0write(4,5) x(i),y(i)write(*,5) x(i), y(i)

5 format (2x,F5.2,2x,F8.3)enddodo i=1,n

write(4,6) x(i), y(i)write(*,6) x(i), y(i)

6 format (2x,F5.2,2x,F8.3)enddoend

function f(u,z)implicit nonereal f,u,zf=u*zreturn

end

8/7/2019 PC: Euler & Runge - Kutta

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5

i

( )c

x

exycx

y

xdxydyxdx

ydy

xydxdyxydx

dyxyy

+

=+=

==

===

2

22

2ln

0>y

( )c

x

exy+

= 22

( ) ey =0

( )

( )1

2

2

0

2

2

10

+

+

=

===

x

c

exy

ceey

ii

, , ( )xfy = Runge Kutta ( )xfy = Euler.