# A Numerical Solution of the Heat Transfer Problem

**A Numerical Solution of the Heat Transfer Problem**

In this master thesis, there is analysis, modeling and a numerical solution of three heat transfer problems. The first problem describes the heat transfer of a two-dimensional composite plate subject to thermal conductivity and thermal convection phenomena. The second problem concerns the heat transfer of a three-dimensional titanium cube in which there is heat generation and subjected to thermal conduction phenomena. Both problems were discretized using finite difference methods: (a) backward differences, (b) Crank-Nicolson. Timing steps were grouped to increase parallel computational work and reduce the communications. The numerical solution is carried out by preconditioned iterative methods based on domain decomposition methods by semi-aggregation techniques in space and time. The proposed scheme showed a significant improvement compared to solving a time step per iteration. The third problem concerns the study of heat transfer through natural convection in a closed rectangular cavity. Modeling led us to the non-dimensional Navier-Stokes equation, the energy equation, and the continuity equation. The solution was performed with the Chorin projection method using finite differences for the discretization of the linear and non-linear terms on a staggered grid. Solving the linear systems in each iteration was performed using a domain decomposition method based on semi-aggregation techniques with satisfactory strong scalability. Finally, the numerical results for the three problems are presented.