The course focuses on the solution of 1D and 2D steady-state and transient heat conduction problems. These were chosen both as one of the fundamental classes of problems in energy applications and as a paradigm for the general case of elliptic and parabolic partial differential equations (PDE). In order to solve this main task, which is addressed using both the finite difference and the finite element approach, computational methods for the solution of several classical problems of numerical analysis are needed. Those methods are presented during the course, as their need naturally arises in the solution of the main task. The fundamental tool for their practical development and for the solution of 1D heat conduction problems is Matlab, to which the initial part of the lab classes is devoted. The chosen tool for the solution of 2D heat conduction problems is the Freefem++ freeware, to which the final part of the lab classes is devoted.

Through this course the student is expected to acquire first of all a feeling of the importance of computation vs. contemplation of different types of analytically intractable equations, a good knowledge of the methods for their solution, the ability to implement and solve them using Matlab and Freefem++, as well as the ability to quantitatively assess the accuracy of the results obtained on the computer (quality assurance).

The knowledge acquired in the following BSc courses (or equivalent ones) will be needed: Calculus (Analisi matematica I e II), Computer science (Informatica), Applied thermodynamics and heat transfer (Termodinamica applicata e trasmissione del calore).

A crash course on Matlab (computational lab)

The 1D steady-state heat conduction problem
• Interpolation concepts and finite difference approximation of derivatives
• Imposing boundary conditions
• Algebraic approximation of the original ordinary differential equation (ODE)
• Methods for the solution of large sets of linear algebraic equations: concept of computational cost; direct vs. iterative methods; concept of condition number.
• Concepts of accuracy and convergence

The 1D transient heat conduction problem
• Fundamental solution of the heat conduction problem
• The method of lines as a general approach to the solution of initial-boundary value PDEs
• Methods for the finite difference solution of initial value problems (single ODE, set of ODEs): Adams-Bashfort, Adams-Moulton, Runge-Kutta, Predictor-Corrector, Backward differentiation formulae for stiff problems.
• Concept of stability. The Lax equivalence theorem. Von Neumann analysis.
• Methods for the solution of nonlinear algebraic problems

The 2D heat conduction problem
• Weak formulation
• Imposing boundary conditions
• Finite element vs. finite difference approximation
• Concepts of mesh generation/triangulation
• Quadrature formulae
• A crash course on Freefem++ (computational lab) Solution of steady state and transient 2D heat conduction problems

40 hours of computational lab are foreseen, where the students will individually work on PCs. Crash courses on Matlab and Freefem++ are foreseen, to allow the students to solve problems of increasing complexity, under the guidance of the teaching assistants.

Selected chapters from:
- A. Quarteroni, R. Sacco, F. Saleri, "Scientific Computing with MATLAB and Octave" 4th ed. (Springer, 2014)
- C. Johnson, "Numerical solutions of PDEs by the finite element method" (Dover, 2009)
- Freefem++ user manual.

The assessment is made by written exam only. Each student works on a PC in the lab and is asked to: 1) solve different numerical problems, using Matlab and/or Freefem++, and summarizing the results in the form of suitable plots; 2) justify the choice of the methods used for the solution; 3) discuss the quality/accuracy of the obtained numerical solution. These three items, collected by the student in a short report (doc file), contribute as follows to the final grade: 1) 60%; 2) 20%; 3) 20%.