Mathematical Modelling of Processes of Spontaneous Ignition in a Stockpile with a Circular and Semicircular Section by Rothe’s and Two-Sided Approximations Methods

Self-ignition of a stockpile of materials such as peat, coal, and grain occurs due to the accumulation of heat released by an exothermic oxidation reaction, so the stockpile can be considered as a body with an internal heat source. The research of self-ignition processes using mathematical modeling is reduced to the need to find a solution to the initial boundary value problem for the two-dimensional semi-linear heat conduction equation. Since it is not always possible to find an analytical solution, it makes sense to use numerical analysis methods. The aim of this article is a numerical study of the initial boundary value problem for a two-dimensional semilinear heat conduction equation that arising in the mathematical modelling of self-ignition processes of a stockpile of bulk material of cylindrical shape with a circular and semicircular base using the Rothe’s method in combination with the method of two-sided approximations based on the Green's function. To achieve this goal, the original initial boundary value problem for the semilinear heat conduction equation using the Rothe’s method was replaced by a sequence of boundary value problems for the semilinear elliptic equation with the Helmholtz operator, each of which was reduced to the nonlinear Hammerstein integral equation. An iterative process of the two-sided approximations method was constructed for it with a stopping condition obtained through a posteriori error estimation. The power of the internal heat source was approximated using an exponential dependence. The results of the conducted computational experiment are presented in the form of graphs of approximations to the solution on different time layers and graphs of heat maps. The resulting approximate solutions for a stockpile of cylindrical shape with a circular and semicircular base were compared with each other and with a known solution for a stockpile of cylindrical shape with a rectangular base.