Help about the numerical computation on the following equations

[PRB147]

Dear every memeber, I encountered the numerical computation on the following
differential equations. I use four-step Runge-Kutta methods to integrate, but
the result is not converge, even if I take the time step (t=0.0001).
please help me why so? Could you please tellme which Fortran subroutine can do the tasks? (I know the following equations has exact solutions.)

$$i\frac{dy_1}{dt}=20.0y_2; \\ i\frac{dy_2}{dt}=20.0y_1$$

 

[HallsofIvy]

What do you mean by "converge"? Runge-Kutta gives a specific result, there is no "convegence" involved. Or do you mean "converge, as the time step gets smaller, to the answer I get by integrating directly"?

 

[J77]

Why the "i"?

 

[PRB147]

"i" is the imaginary unit.
I tried Runge-Kutta in IMSL, and code in the well-known book 'numerical recipe'.
They can not give the correct result. The original equations are 2500 differential equations looking like this
$$i\frac{dp_{ij}}{dt}=-50*(p_{i+1,j}+P_{i-1,j})+50*(p_{i,j-1}+P_{i,j+1})))-ip_{ij}/5$$
i,j=1,2,..50.

Please help me, Dudes.

I can not obtained the correct numerical result.

 

[timur]

Does not look very bad. I think you should first write the equations in the form that does not involve i-s.

 

[PRB147]

Thank you, timur, J77 , Hallsofivy

+50 should be -50.
The problem is that it is high oscillatory, I can obtained the numerical result when
i=1,2,3, and j=1,2,3. when i is larger than 3, the numerical result start to diverge.
That is very tricky

the correct equations should be

$$i\frac{dp_{m,n}}{dt}=-50\times[(p_{m-1,n}+p_{m+1,n})+(p_{m,n-1}+p_{m,n+1})]-ip_{mn}/5$$

i is the the imaginary unit

 

[PRB147]

I solved it.