Solution one-dimensional problems of heat conduction with convection using Poisson integral

Abstract

There is the relevant problem of developing high-viscosity oil in the oil and gas industry. One of the ways increasing its production is the thermal method. It is necessary to combine hydrodynamic equations with the thermal conductivity equation for simulation of filtration processes. The resulting mathematical models are solved by using different numerical methods. The accuracy and convergence of such algorithms is rarely tested. One way to carry out this check is to simulate problems that can be solved analytically in particular cases. For example, Fourier series are used for simple boundary conditions. Another method is usage the Poisson integral. This method is more convenient for comparing results than the Fourier method, because with the same accuracy requires significantly less calculations. But the Poisson method requires knowledge of the initial condition over the entire space, and in real problems it is usually given in a limited area. In this paper we propose an algorithm for extending the initial conditions to the entire space. In this way, two heat conduction problems are solved taking into account convection using the Poisson integral. It is shown that the accuracy is comparable to the Fourier method, but with fewer calculations.