A new approach for solving the time-dependent two-center Dirac equation is presented. The method is based on using the finite basis set of cubic Hermite splines on a two-dimensional lattice. The Dirac equation is treated in the rotating reference frame. The collision of U92+ (as a projectile) and U91+ (as a target) is considered at energy E lab = 6 MeV u−1. The charge transfer probabilities are calculated for different values of the impact parameter. The results obtained are compared with previous calculations (Tupitsyn et al 2010 Phys. Rev. A 82 042701), where a method based on atomic-like Dirac–Sturm orbitals was employed. This work can provide a new tool for the investigation of quantum electrodynamics effects in heavy-ion collisions near the supercritical regime.