Convergence orders

How fast each order of the regime-switching expansion approaches the numerical solution.

Each line is the absolute error of an approximation to the survival probability at a fixed maturity, against the numerical solution, as the switching rate $\lambda$ runs from 1 to 1000. Both axes are logarithmic, so an error of order $\lambda^{-n}$ is a straight line of slope $-n$. The parameters are those of the survival demo.

The averaged, first-order and second-order errors fall with slopes $-1$, $-2$ and $-3$, measured over the last decade of $\lambda$ and reported below the chart. A second-order slope visibly shallower than $-3$ at large $\lambda$ would refute the expansion. At small $\lambda$ the lines bend, because the expansion is asymptotic.

Errors below about $10^{-12}$ reach the accuracy of the in-browser ODE solver, so the plot stops at $\lambda = 1000$.