Numerical Diagnostics of Solution Blow-Up in a Thermoelectric Semiconductor Model

A system of equations with nonlinearity in the electric field potential and temperature is proposed for describing the heating of semiconductor elements on an electrical board with thermal and electrical breakdowns possibly arising over time. A method for numerical diagnostics of solution blow-up is considered. In the numerical analysis of the problem, the original system of partial differential equations is reduced to a differential-algebraic system, which is solved using a single-stage Rosenbrock scheme with complex coefficients. The blow-up of the exact solution is detected using an asymptotically sharp a posteriori error estimate obtained by computing approximate solutions on sequentially refined grids. The blow-up time is numerically estimated in the case of various initial conditions.