Nonlinear Transient Heat Conduction in Multilayer Slabs: Implicit Euler Time Discretization and Finite Difference Method with Newton Linearization

This paper presents a numerical method for solving transient one-dimensional heat conduction problems in multilayer slabs with temperature-dependent thermal conductivities. The governing nonlinear partial differential equations are formulated separately in each layer, allowing for distinct material properties. Perfect thermal contact at internal interfaces is enforced through continuity of temperature and heat flux, while general boundary conditions are imposed at the external boundaries, including prescribed temperature, specified heat flux, and convective exchange. A key feature of the proposed approach is to discretize the partial differential equations first in time using the implicit Euler method, thereby reducing the original problem to a sequence of nonlinear two-point boundary value problems with interface (transmission) conditions. A second-order finite difference scheme is employed for spatial discretization, and the resulting system is expressed in global form using a unified indexing strategy. The system is solved at each time step by Newton linearization, yielding a sparse Jacobian matrix that is tridiagonal in the interior and locally extended at the interfaces. Efficient banded solvers lead to O(N) cost per time step, where N is the number of spatial nodes. Numerical experiments confirm the expected accuracy, unconditional stability, and computational complexity of the method.