Elements of Numerical Analysis

Module MODULO II

Primary objective of the course of Numerical Analysis is to provide students with the concepts and fundamental tools in the study of methods for the numerical solution (i.e. on the computer) of  mathematical models governed by systems of differential equations. The first module is primarily concerned with methods for ordinary differential equations. The second module is an introduction to the methods for the numerical solution of partial differential equation, with particular reference to the equations Mathematical Physics: parabolic, elliptic and hyperbolic equations. Students are exposed to the fundamental notions of consistency, stability and convergence of the methods as well as practical issues that affect their accuracy, efficiency and robustness. For completeness, during the course  the main mathematical properties of these equations are briefly recalled, and some of their main applications to the description of stationary and time-dependent phenomena are illustrated.

Natural continuation of the first module, it is suited for those who have interest in applications of mathematics to a wide variety of real-world models. Anyone wishing to explore the topics covered in the course will then follow the course of Computational Fluid Dynamics, available during the second year of the Master, dedicated to techniques for the numerical solution of the Euler and Navier-Stokes equations that govern the motion of fluids and gases.

The course consists of lectures, during which the various topics are illustrated. Practical sessions will be performed with computer implementation of the main methods explained in class. The exam consists of an oral examination.

The lessons will be face-to-face, or in mixed or remote mode, depending on what is allowed by the containment measures of the pandemic, and in compliance with the safety of the students and the teacher. If necessary, the exam itself wil be done remotely. 

Basic notions of models governed by partial differential equations: Poisson, heat and wave equations.

The notion of wellposedness of problems for the differential equations of mathematical physics.

Heat equation. Recall of some procedures to obtain exact solutions in particular cases: Fourier method and separation of variables method.

Forward Euler method. Stability analysis: von Neuman method. Implicit methods: Backward Euler and Crank-Nicholson scheme. Tridiagonal systems. Heat equations with variable coefficients. Consistency, convergence and stability of finite difference methods for initial value problems. Lax equivalence theorem (statement). Heat equation in multiple dimensions. Fractional step methods. Alternate Direction Implicit (ADI) methods.

Elliptic equations. Finite difference method for the Poisson equation on Cartesian grids. Vertex-center and cell-center discretization. The boundary conditions problem (Dirichlet and Neumann conditions). Level set and ghost point methods for the treatment of arbitrary geometries. Multigrid method for the resolution of the related sparse algebraic system (notes).

Hyperbolic equations. Single linear scalar equation. The method of characteristics. Linear wave equation with constant coefficients in one spatial dimension: solution of the initial value problem.

Finite difference methods for the scalar equation in one spatial dimension. Three-point methods: upwind, Lax-Friedrichs and Lax-Wendroff. Consistency and stability. Courant-Friedrichs-Lewy condition and domain of data dependence.

First-order and second-order methods. Modified equation, dissipation and dispersion. Burgers equation. Method of characteristics. Discontinuous solutions.

Wave propagation in media with time-dependent velocity. Finite difference methods and pseudo-spectral methods Reflection and transmission of waves.

Wave equation in two spatial dimensions. Finite difference method. Consistency and stability. Propagation of a wave packet in two dimensions. Numerical study of the diffraction of a packet. Propagation of a wave packet in media with time-dependent propagation velocity. Connection with geometric optics.

Dispersive equations. Korteweg-de Vries equation. Solitons. Analytical expression and comparison with the numerical solution. Numerical calculation of solitons for quasilinear dispersive equations.

If time allows it, the last topic covered will be the numerical solution of the time-dependent Schroedinger equation for a quantum particle in a potential well.

In addition to the topics listed above, during the course there will be exercises in Matlab that illustrate the implementation of some basic methods.

Please note. If the teaching is taught in mixed mode or remotely, the necessary changes may be introduced with respect to what was previously declared, in order to respect the expected program and reported in the syllabus.

Randall Le Veque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007.

A single book for the transaction of finite difference methods for both ordinary differenzialo partial differential equations. Some topics on the EDP are from this text.

John Strickwerda, Finite Difference Schemes and Partial Differential Equations Paperback – September 30, 2007.

Excellent introductory text on finite difference methods for partial differential equations.

Robert D. Richtmyer, K. W. Morton, Difference methods for initial-value problems, Interscience Publishers, 1967 - 405 pages

A classic text, still valuable for many basic concepts
 

K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, An Introduction, University of Oxford, UK, Second Edition

An introduction to numerical methods (mainly finite difference) for the differential equations of mathematical physics.

The exam is recorded together with the Numerical Analysis module I as a single 12-credit exam.
Each module consists of an oral interview carried out after the end of each course.
The midterm exam of the 12-credit course consists in passing the module of Numerical Methods for Ordinary Differential Equations.
It is at the student's discretion to take the two modules together or separately.

Learning assessment may also be carried out on line, should the conditions require it.