I have solved system of ODEs multiple times in the past. But for some reason this particular problem has made things difficult for me. See Eq.(7) of the attached file and its subsequent discussion. If you can reproduce the results reported in this paper, you will probably be able to help me. I could then share my fortran code with you and seek your help in debugging it. I am actually solving a system of two nonlinear ODES which are of third-order and also coupled.
Is there anyone with an expertise in computational physics and fortran/python/matlab around?
Well, isn't that a classic problem of higher order system of differential equations? Depending on the type of the system, convergence can be very difficult. I don't think there is a "fix it all" approach. Maybe what you could try is get some code that already has a good solver and modify the field equations and see if that converges? I'm thinking on the lines of OpenFOAM or maybe something in Matlab if you use it. And then once you see that your problem solves, you would know what is required in your Fortran code to make it happen?
As mentioned earlier, I have used the regular RK4 or RK5 order algorithms to solve many standard problems before. This time for this problem my code starts giving NaN results after few runs. There are so many parameters in the problem that it is difficult to check every intermediate step. Also I am not sure if the ode solver of Matlab or python have some sort of anything "extra" in their algorithms that is not mentioned in the ode algorithms of standard textbooks.
In Matlab I think you can always look up what the built-in functions do, but as far as I know, it is just the textbook function, and no additional stabilisation or anything. I would recommend to simplify the physics of the problem until it solves, and gradually add in the more complex parts, and then see where the error creeps in, so at least you know what causes them, and then maybe you can see how to fix it.
Here is another possibility. Specialized topics in physics and engineering utilize extremely sophisticated software (sometimes it is commercial software that you have to pay for) that is designed to solve problems such as yours. The same software can also be used for other applications if you know how to enter the inputs to make the problem become your problem. I once used software, designed to simulate an electronic device, to get numerical solutions to Laplace's equation. See if you can find some software that is already available and that was designed for problems resembling your problem.
I only will try to give you some general advises. 1. It seems that the derivatives much more than R_i. Thus the solution should be fast changing or fast oscillating functions. It is very convenient to find the fast oscillating part of the solution, if it is possible. Try to use the solution in the form FG where А is the fast oscillating function and G is a slow function. If You can find А the eq for G will be better for solving numerically. Also, the solution can depend on sqrt(H(R_i,t) or the analogous. If H=0 at some t^* you have a big problem with RC4 or RC5. It is due to the fact that d/dt(sqrt())~1/sqrt(()). You can use the Euler method and Eitken rule. Or simply Bulirsha - Shtera method. I hope the last variant ( Bulirsha - Shtera method) not the best way, but it should work.
Dear Vikash Pandey
Did you compare your result with result obtained by Piecewise analytic method.
Piecewise analytic method (PAM)-(Published January 2012 by T. Abassy) is a new technique used for solving nonlinear differential equation. The advantages of PAM are( these points are proofed):
1. It gives a general analytic formula that can be used in differentiation and integration.
2. It can solve highly non-linear differential equation.
3. The accuracy and error can be controlled according to our needs very easily.
4. It can solve problems which other famous techniques can’t solve.
5. In some cases, it gives the exact solution.
You can follow the updates and discussion and comments on the URL:
https://www.researchgate.net/post/What_do_you_know_about_Piecewise_Analytic_Method_PAM-Published_January_2012_by_T_Abassy
will contain all advantages and disadvantages of PAM.
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