NUMERICAL ANALYSIS

Learning Objectives

• Module 1

  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.

• Numerical Analysis - Second module

  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.

Detailed Course Content

• Module 1

  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.

• Numerical Analysis - Second module

  Basic notion on models governed by partial differential equations: Poisson, heat and wave equation.
  The notion of wellposedness of problems for the differential equations of mathematical physics.

  Heat equation. Exact solutions in particular cases: separation of variables and the Fourier method.
  Forward Euler method. Stability analysis: Von Neuman method. Implicit methods: Euler and Crank-Nicholson schemes. Solution of tridiagonal systems. Heat equation 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) method.

  Elliptic equations. Finite difference method for the Poisson equation on Cartesian grids. Vertex-center and cell-center discretization. The problem of boundary conditions (Dirichlet and Neumann conditions). Ghost level set Methods for the treatment of arbitrary geometry. Multigrid method to solve the relative sparse algebraic linear system (notes).

  Hyperbolic systems. Single scalar linear equation. Finite difference methods. Consistency and stability. Courant-Friedrichs-Lewy condition and domain of dependence on the data. Method of Lax-Friedrichs. upwind methods. First order and second order methods. Modified equation, dissipation and dispersion. Burgers' equation. Features of the method. Weak solutions. viscosity solutions and the entropy principle (notes). conservative methods. Conditions in L1 stability.

  In addition to the items listed above, during the course there will be shown exercises in Matlab or Python (using numpy) that illustrate the implementation of some basic methods.

Textbook Information

• Module 1

  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.

• Numerical Analysis - Second module

  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.