When faced with a new circuit, the software first tries to find a steady state solution wherein all the nodes conform to Kirchhoff's Current Law and the voltages across and through each element of the circuit conform to the voltage/current equations governing that element.
Once the steady state solution is found, the software can analyze the response to perturbations using piecewise approximation, harmonic balance or other methods.
Software such as the PLECS interface to Simulink uses piecewise linear approximation of the equations governing the elements of a circuit.
Every time a diode switches from on to off or vice versa, the configuration of the linear network changes.
More complex digital circuits are typically designed with the aid of computer software.