DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCES
Master's Degree in Mathematics
Academic Year 2019/2020 - 1° Year - APPLIED Curriculum

ISTITUZIONI DI ANALISI NUMERICA

Teaching Staff

Learning Objectives

• MODULO I
  

  The goal of this first part of the course is to introduce the student to the computational issues of the solutions of ordinary differential equations (ODEs) and to give several tools for the numerical resolutions of these problems. In particular,some important concepts will be introduced as: consistency, stabilty and convergence of the nuemerical methods presented during the course. Furthermore some other interesting property of such methods as accuracy and efficiency will be studied. Finally, several Matlab codes regarding the different parts of the course will be presented to the students.

• MODULO II
  

  The object of the couse Numerical methods for partial differential equations" is to introduce the numericla methods for the numericla solution of partial differential equations, in particular for hyperbolic, elliptic and parabolic equations.

Course Structure

• MODULO I
  

  The course of Numerical Analysis is mainly focused on theoretical lessons. During the lessons the theoretical discussion is supported by exercises where applied problems are presented and solved with the help of the computer and by implementing in MatLab of the methods introduced during the lesson.

• MODULO II
  

  Heat equation in one and multiple dimension. Elliptic equation, Poisson equation. Hyperbolic equations. Finite difference methods.

Detailed Course Content

• MODULO I
  

  Initial value problems. A brief introduction to ordinary differential equations (ODEs), existence and

  Uniqueness theorem. Well-posed problem.

  Numerical methods for ODEs: Explicit and Implicit Euler methods, modified Euler method, Heun method. One-step methods, Taylor methods, Runge-Kutta (RK) methods.

  Consistency and convergence of R-K metohds and order conditions. Implicit R-K methods. Variable step size control. Stability analysis for Runge-Kutta methods: stability function, A-stability and L-stability. Stiff problem. Existence and uniqueness of a solution for implicit R-K methods. Collocation methods.

  Multistep methods: Adams, Backward Differentiation Formulas (BDF) and linear multistep methods (LMM), predictor and corrector methods, 0-stability, convergence and consistency of LMM.

  A brief introduction to Differential Differential-Algebraic Equations (DAEs). Definition of differential index and special forms of DAEs. Numerical methods for DAEs. Singular perturbation problems. Partitioned and additive Runge-Kutta methods.

  Boundary value problems (BVPs). Shooting method. Multiple Shooting method. Finite Difference methods (FD) for BVPs. Consistency, stability and convergence.

• MODULO II
  

  Poisson equation, heat equation and waves equation.

  Stability Analysis: Von Neuman method.

  Forward Euler method, Crank-Nicholson method.

  Consistency, convergence and stability for finite difference methods for initial value problems. Lax Theorem.

  Heat equation in multiple dimensions. Alternate Direction Implicit (ADI) method.

  Elliptic Equation. Finite difference method for the Poisson equation on cartesian grid.

  Level set methods and ghost point for arbitrary geometry. Multigrid method.

  Hyperbolic equations. Characteristic method. Finite difference methods: upwind, Lax-Friedrichs e Lax-Wendroff. Consistency and stability. Modified equation. Dispersion and dissipation. Burgers equation.

Textbook Information

• MODULO I
  

  1) G. Naldi, L. Pareschi, G. Russo, Introduzione al calcolo scientifico, McGraw-Hill, 2001.
  Testo semplice ed intuitivo. Capitolo 8 è dedicato ai metodi per la risoluzione di ODE.

  2) A. Quarteroni, R. Sacco, F. Saleri: Matematica Numerica, Springer Italia, 3° Edizione.
  Testo molto ampio e ricco di esempi. Contiene molto materiale e riporta esempi didattici implementati in matlab.

  3)V. Comincioli, Analisi Numerica: metodi, modelli, applicazioni, McGraw-Hill, Milano, 1990.
  Classico testo di Analilsi Numerica, molto vasto. Contiene molto materiale. Utile strumento di consultazione per alcuni argomenti (es. differenze finite o introduzione ai metodi variazioniali).

  4) U. M. Asher e L. R. Petzol, Computer Methods for Ordinary Differential Equations and Differential_Algebraic Equations, Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 1998. Testo utilizzato per la parte riguardante le equazioni differenziali-algebriche.

  5) J. Stoer e R. Bulirsch, Introduction to numerical analysis. Ed. Springer Verlag.

  6) Ernst Hairer, Gerhard Wanner, Syvert P. Nørsett, Solving ordinary differential equations. I. Nonstiff problems. Third edition, Springer, 2008.

  
  7) Ernst Hairer, Gerhard Wanner, Solving ordinary differential equations. I. Stiff problems. Third edition, Springer, 2010.

• MODULO II
  

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

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

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

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