# How does non-dimensionalization improve the behavior of ODE solvers?

I have a set of coupled ODEs that I'm solving numerically. The independent variable is time and runs from values of $$10^{15}$$ to $$10^{17}$$ in units of seconds. The state variables in their usual physical units have large numbers like $$10^{10}$$ and $$10^{50}$$. As these state variables evolve with time, their values systematically increase by ten to twenty orders of magnitude.

I am able to solve my equations with SciPy's solve_ivp (either RK23 or the implicit BDF method) using the default relative error tolerance (rtol=1e-3) and an arbitrary absolute tolerance atol=1. Amazingly, this is with all variables in their default physical units and huge orders of magnitude. However because of the huge scales of my variables, I am wondering if it is worth investing some time in Nondimensionalization.

What advantages would nondimensionalization offer? Would it make the ODE solver behave better in terms of finishing the integration faster but still being precise (i.e., giving the same solution I have now in the dimensionalized version)? Would it provide a natural choice for the atol (and rtol?) parameter which currently I don't really know how to set?

### Background

Let’s have a brief look at what parameters an integrator typically has and what they are used for. All of them govern step-size adaption:

• Relative tolerance (rtol) is usually the most important parameter governing accuracy. After an integration step, the integrator estimates the magnitude of the error it makes and if it exceeds the relative tolerance, a smaller step is taken instead. Values between $$10^{-5}$$ and $$10^{-3}$$ are typical. If you have any way to estimate which relative error would affect your analysis, a few orders of magnitude below this is a good choice. However, usually it’s just how accurate you want your integration to be. Since this value is relative, nondimensionalisation should not affect it.

• Absolute tolerance (atol) is like relative tolerance, just going by absolute values. It is conventionally cumulative with absolute tolerance, i.e., the error needs to be below atol+rtol*state. Unless you set it to a high value or atol to zero, it becomes only relevant if a dynamical variable crosses zero, as relative tolerance cannot cover this well.

• Most integrators have other parameters like the maximum and minimum step size, the initial step size and further parameters governing the step-size adaption.

### Why nondimensionalise?

For a normal integrator, the default values of these parameters are made for a setup where all your dynamical variables have the same order of magnitude, that order of magnitude is 1, and the smallest time scale of your dynamics also has an order of magnitude of 1.

If this does not apply, you have to consider all integration parameters (not just the tolerances) as to whether they need to be adjusted. For many integrators, you also have the problem that you cannot set the absolute tolerance per component. Unless you know what you are doing and can set all the parameters accordingly, it’s easier and safer to nondimensionalise.

Possible consequences of badly set integration parameters include a failing integration, bogus results, and an overly slow integration.

Some thoughts on your specific scenario:

• Your time scales are very high. I would have to look into the details of the integrators you used, but it might very well be that you barely made use of the adaptive integration, and instead progressed with the default maximum time step (which in turn is still very small compared to the time scale of your dynamics) and thus wasted computation time.

• Your absolute error was probably be too low. If your dynamical variables never become zero, it is not surprising if this had no effect. You may also just have been lucky. In general, if you have zero crossings and you have dynamical variables of different orders of magnitude, you cannot have an absolute error that is appropriate for all of them (see above).

• Your relative error is okay.

• If your dynamics varies on several orders of magnitude, consider performing the entire integration in the logarithmic domain.

### Digression: Why can’t the integrator do this?

• Logarithmic time is something that only makes sense if you have a special point in time such as the big bang (in cosmology) or the time of a mutation (in population genetics) and if your timescales span many orders of magnitude (which doesn’t apply to you). Also, you obviously cannot do this, if you want to start at $t=0$. If that is given, I don’t see how this would introduce sensitivity to initial conditions as the dynamics itself should not change. Commented Jan 29, 2023 at 10:36
• Also, what do rtol and atol intuitively mean for dlogX/dlogt with logarithmic state variables, and should atol scale with the initial conditions somehow? – When working logarithmically, you usually only want to work with atol as it effectively gives you relative tolerance. Any rtol would mean that you accept your actual relative error to be larger when the order of magnitude of your variable is larger, which usually makes little sense. Commented Jan 29, 2023 at 10:40
• This is especially confusing if the initial condition is X=1, so dlog(X)=0. What would a good atol be in this case...? – I don’t see a problem with atol here, as zero is not a special point for it. Interpreting rtol may be weird, but see my above comment for that. Commented Jan 29, 2023 at 10:41