A one-dimensional dissipative magnetohydrodynamics code is used to investigate the discharge dynamics of a waveguide for high-intensity laser pulses: the gas-filled capillary discharge waveguide. Simulations are performed for the conditions of a recent experimental measurement of the electron density profile in hydrogen-filled capillaries [D. J. Spence et al., Phys. Rev. E 63, 015401 (R) (2001)], and are found to be in good agreement with those results. The evolution of the discharge in this device is found to be substantially different to that found in Z-pinch capillary discharges, owing to the fact that the plasma pressure is always much higher than the magnetic pressure. Three stages of the capillary discharge are identified. During the last of these the distribution of plasma inside the capillary is determined by the balance between ohmic heating, and cooling due to electron heat conduction. A simple analytical model of the discharge during the final stage is presented, and shown to be in good agreement with the magnetohydrodynamic simulations.