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Numerical simulation models are utilized for the development and the design of semiconductor devices to a steadily growing extent. However, the simulation programs to date are known only to be handled under some restrictions. This paper presents the novel program system BAMBI, capable of simulating the two-dimensional transient behavior of arbitrarily shaped devices. The exact numerical model accomplishes locally refined grid structures with automatic setup and adaption to the different stationary and transient operating conditions. An automatic time step control algorithm has been implemented and a "moving" grid is introduced. The program system is applied to the simulation of power MOSFET's. Calculated output characteristics including the resistance, saturation, and carrier multiplication region are discussed. Special emphasis is placed on the avalanche generation process and the analysis of the turn on of the device.