The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.
Abstract— The fundamental possibility of a numerical solution of the time-dependent Schrödinger equation for a wave function defined on a non-uniform coordinate grid by the spectral method using the fast Fourier transform algorithm is discussed. The method is based on reducing the non-uniform coordinate grid to a uniform one by a non-linear transformation of coordinates and approximating the obtained evolution operator using the Lie–Trotter–Suzuki product formula. Algorithms for the numerical solution of the first and second orders are developed.