My problem involves the solution of a second-order ODE with a fixed-step (input and output). Specifically, this ODE is the radial part of Dirac and Schrödinger equation for a spherical symmetric potential.
This is for example the ODE of the radial part Schrödinger equation:
$$\left(\frac{d^2}{dr^2} +2r\frac{d}{dr} -\frac{l(l+1)}{r^2} \right) R(r)+V(r)P(r)=E\,R(r)$$
or by substituing R(r) = P(r)*r:
$$\left(\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2} \right) P(r)+V(r)P(r)=E\,P(r)$$
$V(r)$ is defined on a fixed grid, that why I need a fixed-step solver. $E$, $l$ are numerical predefined values. $r$ can have values in the interval from $[0,\infty]$, but usually it is only interesting up to an certain $r_{max}$. The initial condition are $P(0) = 0$ and $P'(0) = 0$.
Commonly used solvers are multistep methods like Adams–Bashfort or also extrapolation methods like the GBS method. I already have some implementation exactly using these aforementioned methods, which were specifically created to deal with this certain problem.
For comparison, I am searching for a software package or library, which has some evolved fixed-step solver implemented.
Are there libraries that contain general-purpose fixed-step ODE solvers?
