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A rotating superfluid forms an array of quantized vortex lines which determine its angular velocity. The spasmodic evolution of the array under the influence of deceleration, dissipation, and pinning forces is thought to be responsible for the phenomenon of pulsar glitches, sudden jumps in the spin frequency of rotating neutron stars. We describe and implement an $N$-body method for simulating the motion of up to 5000 vortices in two dimensions and present the results of numerical experiments validating the method, including stability of a vortex ring and dissipative formation of an Abrikosov array. Vortex avalanches occur routinely in the simulations, when chains of unpinning events are triggered collectively by vortex-vortex repulsion, consistent with previous, smaller-scale studies using the Gross-Pitaevskii equation. The probability density functions of the avalanche sizes and waiting times are consistent with both exponential and log-normal distributions. We find weak correlations between glitch sizes and waiting times, consistent with astronomical data and meta-models of pulsar glitch activity as a state-dependent Poisson process or a Brownian stress-accumulation process, and inconsistent with a threshold-triggered stress-release model with a single, global stress reservoir. The spatial distribution of the effective stress within the simulation volume is analysed before and after a glitch.