A computational algorithm for constructing a two-dimensional heat wave generated by a non-stationary boundary condition

• Published in: Journal of physics. Conference series Vol. 1392; no. 1; pp. 12083 - 12088
• Main Authors: Kazakov, A L, Spevak, L F, Lempert, A A, Nefedova, O A
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 01.11.2019
• Subjects: Algorithms; Boundary conditions; Boundary value problems; Degeneration; Numerical analysis; Physics; Poisson equation; Polar coordinates; Self-similarity; Thermodynamics; Wave fronts; Wave propagation
• ISSN: 1742-6588; 1742-6596; 1742-6596
• DOI: 10.1088/1742-6596/1392/1/012083

The paper discusses solutions of the nonlinear heat equation, which have the form of a heat wave propagating on a zero background with a finite velocity. Such solutions are not typical for parabolic equations, and their existence is associated with the degeneration of the problem at the wave (zero) front. We propose a numerical algorithm for constructing a two-dimensional heat wave, symmetrical with respect to the origin, with a non-zero boundary condition defined on the moving boundary. The main difficulty of the new task is that at each time point a heat wave front (a domain boundary) is unknown. The solution is carried out in two stages. At first, we change the roles of unknown function and radial polar coordinate. For a new unknown function at each time point, we obtain a boundary value problem for the Poisson equation in a known region. The step-by-step solving of this problem by the method of boundary elements at a given time interval allows us to determine the law of the zero front moving. At second, we approximate the found zero front by an analytical function and construct a generalized self-similar solution. The developed algorithm is implemented and tested on a task set.