In turbomachines, heat fluxes are generated during unilateral contact and frictional rotor-stator interactions. Though they are expected to play a significant role in the overall dynamics of engines, their precise influence in the dynamics is not well understood. This contribution proposes a blade-casing reduced-order thermomechanical model involving unilateral contact and frictional terms formulated as a Measure Differential Inclusion together with various numerical methods able to properly capture the dynamics. Comparison with less advanced explicit time-marching schemes is provided. It is shown that the MDI discretized via an implicit scheme is more robust. A sensitivity analysis to the thermal coupling and model reduction is provided. It is found that the proposed solution method sets a sound basis for efficient numerical simulations of thermomechanical rub occurrences in turbomachinery.