Numerical Solutions for a Coupled Parabolic Equations Arising Induction Heating Processes

In this paper we study the numerical solution for a coupled para-bolic equations. The system is derived from an induction heating process. An implicit finite-difference scheme for a coupled parabolic system is proposed and analyzed. Some numerical experiments are performed. We found that the numerical solutions do match the theoretical results obtained from the previous study. Moreover, some numerical results show new phenomenon which has not been proved up to now. 1. Introduction. Induction heating is a common method used in modern industry to bond, harden or soften metals or other conductive materials (see [1]). The advantage of this method combines speed, consistency and controllability. The investigation of an induction heating system usually relies on a series of expensive, time-consuming and complicated experiments. The mathematical analysis and numerical simulation for induction heating play an important role in the designing process. The induction heating process can be modeled by Maxwell's equations coupled with a nonlinear heat equation (see [2, 3, 6, 7]). It is quite complicated to analyze the full system when the electric conductivity depends on the temperature. In the previous study of induction heating, a common method is to assume that the electric field E is given by certain special time-harmonic form and then to decouple Maxwell's equations from the nonlinear heat equation (see [1, 3, 4] for examples). This method provides a good approximation of the full system when the electrical conductivity of the targeted material is not sensitive with respect to the change of the temperature. However, if the electric conductivity strongly depends on the temperature, one has to take into account the effect of the temperature and investigate the full Maxwell equations coupled with the nonlinear heat equation. The investigation has been carried out in several papers (see [6, 7, 8] for examples). This paper is devoted to finding the numerical solutions for the coupled system. As a first step, we only study the one-dimensional case and the three-dimensional case will be investigated in a separated paper. The paper is organized as follows. In section 2, we describe the mathematical model and present the theoretical results. In section 3, the numerical method is