Matrix-implicit Krylov-subspace methods have made it possible to efficiently compute the periodic steady-state of large circuits using either the time-domain shooting-Newton method or the frequencydomain harmonic balance method. However, the harmonic balance methods are not so efficient at computing steady-state solutions with rapid transitions, and the low-order integration methods typically used with shooting-Newton methods are not so efficient when high accuracy is required. In this paper we describe a Time-Mapped Harmonic Balance method (TMHB), a fast Krylovsubspace spectral method that overcomes the inefficiency of standard harmonic balance in the case of rapid transitions. TMHB features a non-uniform grid to resolve the sharp features in the signals. Results on several examples demonstrate that the TMHB method achieves several orders of magnitude improvement in accuracy compared to the standard harmonic balance method. The TMHB method is also several times faster than the standard harmonic balance method in reaching identical solution accuracy.