Mathematical and Numerical Methods in Engineering

GOALS AND CONTENTS

The didactical goal is twofold. First we intend to present some classical differential models of Continuum Mechanics, developing and analyzing some finite difference schemes for their approximation. Second we want to introduce the variational formulation of some boundary value problems together with the finite element method for their numerical approximation. The course is characterized by a constant synergy between modeling, theoretical aspects and numerical simulation.

TOPICS

Mathematical Methods

Differential calculus for functions of several real variables. Series of functions. Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition. Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in the half-space. Duhamel principle. Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials. String equations. Well-posed problems and separation of variables. D’Alembert formula. Kirchhoff formula and Huygens principle. Lebesgue integral. Projection theorem and Riesz representation theorem. Bilinear form, abstract variational problems and Lax-Milgram lemma. Schwartz distributions. Function spaces of finite energy (Sobolev spaces). Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations.

Numerical Methods

Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability. Approximation with finite differences. Convergence, consistency, zero-stability and absolute stability. Forward Euler-centered scheme. Upwind, Lax-Friedrichs and Lax-Wendroff schemes. Analysis of the schemes, CFL condition and its meaning. Backward Euler-centered scheme. A quick description of systems and of non-linear problems. Discretization with finite differences of a one-dimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation e matrix properties. Diffusion-convection and diffusion-reaction problems. Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis. Discretization of the wave equation with finite difference explicit and implicit schemes. Leapfrog and Newmark schemes. Stability properties.  Introduction to the Galerkin method for a one-dimensional elliptic problem. Consistency, stability and convergence. Cea' Lemma. The finite elements method. Linear and quadratic finite elements. Definition of Lagrangian basis functions, of composite interpolation and error estimates. Algebraic lifting. Extension to the 2D case. Condition number of the stiffness matrix. Approximation of the diffusion-convection-reaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy, strongly consistent methods and mass lumping technique. Approximation of parabolic problems: the semi-discrete problem. Explicit and implicit time marching schemes, the theta-method. Stability properties. A quick description of finite elements for hyperbolic problems. Approximation of the Stokes problem.

BASIC REFERENCES

L. Formaggia, Luca, F. Saleri, A. Veneziani: Applicazioni ed esercizi di modellistica numerica per problemi differenziali, Springer Italia, 2005. A. Quarteroni: Modellistica Numerica per Problemi Differenziali, Springer Italia, 2012. A. Quarteroni, F. Saleri, P. Gervasio: Calcolo Scientifico, Springer Italia, 2012. S. Salsa, F. Vegni, A. Zaretti, P. Zunino: Invito alle Equazioni a Derivate Parziali,  Springer Italia, 2009.

EVALUATION

The exam consists of compulsory two written tests. The test on Numerical Methods also requires the use of a computer. The written tests are followed by two oral examinations: one is compulsory (Mathematical  Methods), the other optional (Numerical Methods).