The paper presents a numerical method for efficiently simulating the time evolution of quantum spin chains and systems under local Hamiltonian interactions. The efficiency of the method depends on the amount of entanglement involved in the simulated evolution. The key idea is to exploit the fact that in one spatial dimension, low-energy quantum dynamics are often only slightly entangled. The method involves an efficient decomposition of the initial state and updating this decomposition when a unitary transformation is applied to one or two nearest neighbor systems. The simulation scheme is based on the Trotter expansion, which approximates the evolution operator by a product of $n$-body transformations. The computational cost grows linearly with the number of systems, making it feasible to simulate time-dependent properties of low-energy dynamics in one-dimensional quantum many-body systems. The method has been tested in various one-dimensional settings, demonstrating its effectiveness in finding approximations to the ground state and simulating time evolutions of local perturbations.The paper presents a numerical method for efficiently simulating the time evolution of quantum spin chains and systems under local Hamiltonian interactions. The efficiency of the method depends on the amount of entanglement involved in the simulated evolution. The key idea is to exploit the fact that in one spatial dimension, low-energy quantum dynamics are often only slightly entangled. The method involves an efficient decomposition of the initial state and updating this decomposition when a unitary transformation is applied to one or two nearest neighbor systems. The simulation scheme is based on the Trotter expansion, which approximates the evolution operator by a product of $n$-body transformations. The computational cost grows linearly with the number of systems, making it feasible to simulate time-dependent properties of low-energy dynamics in one-dimensional quantum many-body systems. The method has been tested in various one-dimensional settings, demonstrating its effectiveness in finding approximations to the ground state and simulating time evolutions of local perturbations.