  A general {\it ab-initio} and non-perturbative method to solve the
time-dependent Schr\"odinger equation (TDSE) for the interaction of a strong
attosecond laser pulse with a general atom, i.e., beyond the models of
quasi-one-electron or quasi-two-electron targets, is described. The field-free
Hamiltonian and the dipole matrices are generated using a flexible $B$-spline
$R$-matrix method. This numerical implementation enables us to construct
term-dependent, non-orthogonal sets of one-electron orbitals for the bound and
continuum electrons. The solution of the TDSE is propagated in time using the
Arnoldi-Lanczos method, which does not require the diagonalization of any large
matrices. The method is illustrated by an application to the multi-photon
excitation and ionization of Ne atoms. Good agreement with $R$-matrix Floquet
calculations for the generalized cross sections for two-photon ionization is
achieved.
