Free-electron lasers (FELs) operate at wavelengths from millimeter waves through hard x-rays. At x-ray wavelengths, FELs typically rely on self-amplified spontaneous emission (SASE). Typical SASE emission contains multiple temporal 'spikes' which limit the longitudinal coherence of the optical output; hence, alternate schemes that improve on the longitudinal coherence of the SASE emission are of interest. In this paper, we consider electron bunches that are shorter than the SASE spike separation. In such cases, the spontaneously generated radiation consists of a single optical pulse with better longitudinal coherence than is found in typical SASE FELs. To investigate this regime, we use two FEL simulation codes. One (MINERVA) uses the slowly-varying envelope approximation (SVEA) which breaks down for extremely short pulses. The second (PUFFIN) is a particle-in-cell simulation code that is considered to be a more complete model of the underlying physics and which is able to simulate very short pulses. We first anchor these codes by showing that there is substantial agreement between the codes in simulation of the SPARC SASE FEL experiment at ENEA Frascati. We then compare the two codes for simulations using electron bunch lengths that are shorter than the SASE slice separation. The comparisons between the two codes for short bunch simulations elucidate the limitations of the SVEA in this regime but indicate that the SVEA can treat short bunches that are comparable to the cooperation length.