The dynamics of one-dimensional, piston-driven hydrogen–air detonations are predicted in the presence of physical mass, momentum and energy diffusion. The calculations are automatically verified by the use of an adaptive wavelet-based computational method which correlates a user-specified error tolerance to the error in the calculations. The predicted frequency of 0.97 MHz for an overdriven pulsating detonation agrees well with the 1.04 MHz frequency observed by Lehr in a shock-induced combustion experiment around a spherical projectile, thus giving a limited validation for the model. A study is performed in which the supporting piston velocity is varied, and the long time behaviour is examined for an initially stoichiometric mixture at 293.15 K and 1 atm. Several distinct propagation behaviours are predicted: a stable detonation, a high-frequency pulsating detonation, a pulsating detonation with two competing modes, a low-frequency pulsating detonation and a propagating detonation with many active frequencies. In the low-frequency pulsating mode, the long time behaviour undergoes a phenomenon similar to period-doubling. Harmonic analysis is used to examine how the frequency of the pulsations evolves as the supporting piston velocity is varied. It is found that the addition of viscosity shifts the neutral stability boundary by about 2 % with respect to the supporting piston velocity. As the supporting piston velocity is lowered, the intrinsic instability grows in strength, and the effect of viscosity is weakened such that the results are indistinguishable from the inviscid predictions.