Numerical solution of the third boundary value problem for the nonlinear mixed heat conduction equation

The paper considers a mathematical model for a mixed nonlinear heat equation with boundary conditions of the third kind. This MM models the process of switching off an electric arc in a co-current gas flow with the addition of a period of stable combustion until the alternating current crosses zero, when the arc is turned off. In this case, the strictly hyperbolic heat equation obtained by the generalized Fourier law is replaced by a hyperbolic-parabolic one. The numerical calculation of the problem is carried out in two stages using an implicit conservative difference scheme, taking into account a variable thermal conductivity coefficient, a nonlinear heat source and a lateral heat sink. At the first quasi-stationary stage, a parabolic equation is considered, in which the thermal relaxation coefficient is equal to zero. Its solution is used to formulate an initial boundary value problem for a hyperbolic equation at the moment the arc is turned off, where the specified coefficient becomes a constant value greater than zero. This second stage implements a significantly non-stationary process of turning off the electric arc.