# Implicit Euler time discretization and FDM with Newton method in nonlinear heat transfer modeling

• 1 Department of Computer Science, University of Chemical Technology and Metallurgy, Bulgaria
• 2 Department of Applied Analysis and Computational Mathematics, MTA-ELTE Research Group, Eötvös Loránd University, Hungary

## Abstract

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.

## References

1. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd Edition, John Wiley & Sons, Inc., (2002)
2. H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, (1986)
3. S.L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover Publications, (1989)
4. J. Lienemann, A. Yousefi, J.G. Korvink, Nonlinear Heat Transfer Modeling, In: P. Benner, D.C. Sorensen, V. Mehrmann (eds) Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, 45 (2005) 327-331
5. U.M. Ascher, S.J. Ruuth, R.J. Spiteri: Implicit-Explicit RungeKutta Methods for Time-Dependent Partial Differential Equations, Appl Numer Math, vol. 25 (2-3) (1997)
6. U.M. Ascher, R.M.M. Mattjeij, R.D. Russel, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, in: Classics in Applied Mathematics, vol. 13, SIAM, (1995)
7. S.M. Filipov, I.D. Gospodinov, I. Faragó, Shooting-projection method for two-point boundary value problems, Appl. Math. Lett. 72 (2017) 10-15 https://doi.org/10.1016/j.aml.2017.04.002