This question was already asked here and this is a suitable form of the equation for numerically solving. \begin{align*}i\frac{\partial}{\partial t}u_{\ell}(r,t) = \Bigg(-\frac{1}{2} \frac{\partial ^2}{\partial r^2} + \frac{\ell(\ell+1)}{2r^2} +  V(r)\Bigg)u_{\ell}(r,t)\quad \quad \quad \quad \quad \quad \quad \nonumber\\  + \frac{2}{3}k(t)r^2 \sum_{\ell' = {max}(\ell-2,0)}^{{min}(\ell+2,L_{{max}})} \Bigg(\delta_{\ell,\ell'} - \sqrt{ \frac{4\pi}{5}} \alpha(\ell, \ell') \Bigg) u_{\ell'}(r,t)\end{align*}where the coefficient $\alpha(\ell, \ell')$ is given by\begin{align*} \alpha(\ell, \ell') = \int Y^*_{\ell0}(\Omega)Y_{20}(\Omega)Y_{\ell'0}(\Omega) d\Omega\end{align*}which the integral can be solved by Wigner-3j coefficients (see  here ).Can anyone suggest an efficient numerically method for solving the equation?
Time-dependent Schrodinger equation(time-dependent Hamiltonian)
numerical analysis;finite difference