What does the exponential function mean in numerical ODE solving formulas?

Question

I'm trying to read papers on numerical ODE algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in New families of symplectic splitting methods for numerical integration in dynamical astronomy there are a few of them. Is it the ordinary exponential that is meant or is it just like a way of writing something else (just like with complex numbers)?

Answer 1

This is a way of writing the flow of a vector field. In other words, if you have an ordinary differential equation (ODE) given by $\dot{x}(t) = v(x(t))$ with initial value $x(0) = x_0$, you could formally write its solution as $x(t) = \mathrm{e}^{t v} x_0$. The formal solution coincides with the actual solution when $v$ is a linear mapping (think of the case in which $v(x) = A x$ for some matrix $A$).

Similarly, if we have another ODE $\dot{y}(t) = w(y(t))$ and an initial value $y(0) = y_0$, then we could write $\mathrm{e}^{t v} \mathrm{e}^{s w} y_0$ to represent the curve resulting from solving the second initial value problem up to time $s$, then using $y(s; y_0)$ as the initial value of the first ODE $\dot{x} = v(x)$ and solving that initial value problem up to time $t$. To reiterate, you can also think about the product of exponentials as meaning that you must first obtain the integral curve of the vector field $w$ that passes through $y_0$, follow said curve until it reaches $y(s)$, and then follow the integral curve of the vector field $v$ that starts at $y(s)$ and goes on for another $t$ units of time. I hope this helps clarify things a little bit more.