Fourier-envelope algorithms are an important component of the mixed-signal/RF verification toolbox. In this paper, we address the unpredictability and lack of robustness that has been reported for these algorithms. We show that the problem stems from fast oscillations in envelopes that are expected to be slowly varying. We demonstrate that this is related to the fact that the envelope equations are always stiff, whether or not the underlying system is. We show that careful choice of envelope initial conditions is necessary to obtain useful solutions, and propose two techniques for finding good initial conditions. Applying these, and solving the envelope equations with stiffly-stable numerical methods, we improve the robustness and reliability of Fourier-envelope methods. We illustrate the new methods with a direct-downconversion mixer circuit.