We present a new concept—the space-dependent Wielandt shift (SDWS)—for accelerating the convergence of the power iteration (PI) scheme for multigroup diffusion k-eigenvalue problems. The SDWS improves on standard Wielandt shift (WS) techniques, which are empirical in nature and are typically effective only when the current estimate of the solution is reasonably converged. By accounting for the physics of the problem through SDWS, we are able to improve the acceleration for the initial iterates when the current estimate of the solution is not close to convergence. Numerical results from one-dimensional problems suggest that, compared to standard WS techniques, the new SDWS techniques can provide upward of a 46% reduction in the number of PIs required for convergence and a 40% reduction in the computational time required. This improvement is sensitive to several problem-dependent factors, such as the geometry and energy-dependence of the problem, the spatial discretization, and the initial guess. The reduction in computational time is also dependent on the linear solver in the PI scheme, as it is well known that WSs can significantly worsen the conditioning of the diffusion linear system. In this paper, we provide a detailed study of the impact of WSs on the performance of several iterative linear solvers. Results from our implementation of SDWS in the three-dimensional (3D) code MPACT show that SDWS can provide similar speedups for 3D multigroup diffusion eigenvalue problems. These results also show that moderate speedups can be obtained by applying SDWS to the coarse mesh finite difference (CMFD) solver in a CMFD-accelerated transport scheme. However, the benefit of doing this may be limited because all but the first few CMFD solves are relatively easy to converge, regardless of the WS used.