The paper presents a new approach to modeling high-current lightning discharges in the Earth's atmosphere, which include compact intracloud discharges and initial breakdown pulses. The model has high spatiotemporal resolution and takes into account the asymmetry between threshold propagation fields of positive and negative streamers, evolution of discharge channels parameters, and the possibility of their simultaneous growth and decay in different parts of discharge tree. The key feature of the model is consideration of discharge channel thermodynamics. This makes it possible to relate the conductivity and radius of the current-carrying part of a hot leader channel to its temperature, which is necessary for reproduction of the currents characteristic of high-current lightning discharges with amplitudes of tens of thousands of amperes. The work is presented in two parts. This article is the first part, which describes modern ideas about high-current lightning discharges and presents a numerical model that is able to reproduce their evolution. Simulation results and their detailed analysis, which allows us to formulate a new development scenario for compact intracloud discharges and initial breakdown pulses, will be presented in the second part of the study.