Solving a system of nonlinear inhomogneous ordinary differential equations?

Hello everyone, I am planning to solve an extremely large nonlinear inhomogeneous ordinary differential equations (20 and more!). The solution should give me (exponential) dynamics of photochemical process (so the whole process is practically finished after few miliseconds).

I have two question here: 1) Is it recommended to use Linearization of the system of ODEs? The stability is definitely a problem, but I think they are more "analytical" than numerical method. Or is it better just to use numerical solvers? I am not very fond of mathematical definition of "stiffness" and so I am very unsure which ODE solver I should use. Since I have no experience in doing either of those, it's really hard to determine which method gives more accurate result.

2) I really have hard time with the input of differential equations on MATLAB. The equation is so large that it takes hours to write them and then check to see if I really got it right. Do you guys have any tip on how to be able to write large equations rather easily?

Thank you.

[John D'Errico]

Writing large systems of equations will take time. If big problems were easy, they would be called small problems, and you would be working on something more interesting.

If you cannot determine if your system is stiff, the ODE solver will tell you, in the sense that ODE45 will probably give up trying to solve it, trying to reduce the time step to a too tiny value. This is the simple test. It usually works quite nicely. If ODE45 does fail, then try a stiff solver. If the stiff solver worked when ODE45 failed, then you have your answer.

From your description, I'd not be amazed if the system was stiff. But stiff essentially means that you have some processes that operate MUCH more quickly than some others in your system, so they have seriously different rates. The ODE45 test is simple to try though.

[Hayao]

Thank you!

I see. I am most likely going to see stiff system then. Several parameters causes some process to be in nanosecond order while other being millisecond order. That is 10^-6 difference and that is huge.

Does ODE45 always fail if the system is too stiff? What should I use first if ODE45 fails (because it's too stiff)? ODE15s? ODE23s? ODE23t? ODE23tb?

[Star Strider]

The ode15s function would be my first choice for a stiff solver.

[John BG]

Hayao, could you please post a concise question, the ODE to solve

[John D'Errico]

I'd expect to see at least some problems in that system with that rate differential. I'm also not at all surprised to hear about such a rate differential.

I agree with Star. ODE15s is the one I've gone too first when ODE45 has failed me, and it has usually worked quite nicely. If you look at

doc ode15s

I see a list of the various solvers, complete with guidelines telling you when to prefer one or another solver.

[Hayao]

Okay, Thank you.

Do you think numerical method is better than linearization of ODEs?