I have a set of initial value problem(IVP) formulated by ODEs.
Usually, if I want to get the solution at some point, numerical integration of the IVP is needed. Also, the sensitivity of the integration with respect to the small perturbations of the initial point can be calculated by the numerical differentiation.
I'm wondering which computation is numerically cheaper in general cases.
Thanks
It depends on the particular problem, how fluctuating the functions are, and the order of error you want in your solutions.
There are some paper on this and it seems that the integration and the generation of derivatives are intertwined, because they both take use of the information of ODEs(i.e. look for key words "internal numerical differentiation"). Which calculation is more time consuming if the same integrator is used and the function is nonlinear. Also the the error of order is the same
Two different things: numerical integration and sensitivity to initial conditions. You could try, as a first approach a leap-frog method or, if you want something a bit more "expensive" a Runge-Kutta-Fehlberg (e.g. 4-5 or 6-7 order). To study the sensitivity to small perturbations I would recommend the integration of the variational problem associated to your IVP. It's a good way to see how perturbations grow. I've done it for the N-Body gravitational problem to study the growth of perturbations, also visible with the Lyapunov exponents.
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