Abstract: Lattice Boltzmann method, as an efficient mesoscopic computational fluid dynamics method, has been developed rapidly in the past two decades. Its relatively high computational efficiency and flexibility make it suitable for the simulation of various complex flows. However, due to its own limitations, the standard lattice Boltzmann method typically utilize uniform rectangular grid, which is not suitable for the simulation of complex flows. Therefore, local grid refinement based on the lattice Boltzmann method has been considered by many researchers. For this purpose, the distribution functions between coarse and fine grids need to be converted at the interface of coarse and fine grids. At present, most conversion methods of distribution function are derived without the presence of the source term, and the previously limited derivation of conversion formulas considering the source term was based on the first-order Chapman-Enskog expansion, which is relatively complex and may limit the application of local grid refinement algorithm in higher-order lattice Boltzmann methods. In this paper, we provide a concise derivation to relate the distribution functions between coarse and fine grids considering an arbitrary source term, based on consistency requirements of the distribution function of the continuous Boltzmann equation between coarse and fine grids. The proposed method is independent of the Chapman-Enskog expansion and Chapman-Enskog approximation, and can be applied to both single relaxation time and multiple relaxation time collision models. In addition, this paper also proves theoretically that the consistency of the first-order Chapman-Enskog approximation of the non-equilibrium distribution between the coarse and fine girds can ensure the consistency of the entire non-equilibrium distribution, which expands the applicability of the previous conversion relationship. Finally, these theoretical results are validated by numerical simulations of a forced Taylor-Green vortex flow, convection-diffusion in a planar Poiseuille flow and lid-driven cavity flow. The good numerical results confirm the adaptability of the conversion relation in the presence of complex source terms and the advantages of local grid refinement technology in dealing with complex flow problems. At the same time, through the simulation of one-dimensional shear wave problem, it is found that the numerical viscosity caused by local grid refinement has a great relationship with the selection of refinement region.