  Lightning discharge is a complicated process with relevant physical scales
spanning many orders of magnitude. In an effort to understand the
electrodynamics of lightning and connect physical properties of the channel to
observed behavior, we construct a simulation of charge and current flow on a
narrow conducting channel embedded in three-dimensional space with the time
domain electric field integral equation, the method of moments, and the
thin-wire approximation. The method includes approximate treatment of
resistance evolution due to lightning channel heating and the corona sheath of
charge surrounding the lightning channel. Focusing our attention on preliminary
breakdown in natural lightning by simulating stepwise channel extension with a
simplified geometry, our simulation reproduces the broad features observed in
data collected with the Huntsville Alabama Marx Meter Array. Some deviations in
pulse shape details are evident, suggesting future work focusing on the
detailed properties of the stepping mechanism.
