Implicit Euler time discretization and FDM with Newton method in nonlinear heat transfer modeling
For engineers modeling heat transfer in materials with temperature-dependent conductivity, this offers an alternative discretization order, but the approach is incremental.
The paper models 1D nonlinear heat transfer by first discretizing in time with implicit Euler, then solving the resulting nonlinear boundary value problems via Newton's method and finite differences. Tested on semiconductor data, it provides a MATLAB implementation.
This paper considers one-dimensional heat transfer in a media with temperature-dependent thermal conductivity. To model the transient behavior of the system, we solve numerically the one-dimensional unsteady heat conduction equation with certain initial and boundary conditions. Contrary to the traditional approach, when the equation is first discretized in space and then in time, we first discretize the equation in time, whereby a sequence of nonlinear two-point boundary value problems is obtained. To carry out the time-discretization, we use the implicit Euler scheme. The second spatial derivative of the temperature is a nonlinear function of the temperature and the temperature gradient. We derive expressions for the partial derivatives of this nonlinear function. They are needed for the implementation of the Newton method. Then, we apply the finite difference method and solve the obtained nonlinear systems by Newton method. The approach is tested on real physical data for the dependence of the thermal conductivity on temperature in semiconductors. A MATLAB code is presented.