A general {it ab-initio} and non-perturbative method to solve the time-dependent Schrodinger 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.
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