Numerical investigation of the nonlinear heat diffusion equation with high nonlinearity on the boundary

A nonlinear heat diffusion problem is considered when the thermal conductivity and heat capacity are nonlinear functions of the temperature. At one of the boundaries a highly nonlinear condition is imposed involving both the flux and the temperature. We apply equivalence transformation which allows to reformulate the problem as an equation with linear diffusion for the transformed function. This gives a unique opportunity to create a specialized implicit finite difference scheme with internal iterations that faithfully represents the energy balance for the system. The equivalence transformation allows one to treat problems with plane, cylindrical, and spherical symmetry in an unified fashion. As a featuring example we consider two versions of the nonlinear boundary condition: energy absorption and energy input. We show that the latter leads to blow up of the solution at the boundary and identify the profile of the blowing-up solution.