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Numerical methods for partial differential equations

Academic Year 2023/2024 - Teacher: Giovanni RUSSO

Expected Learning Outcomes

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.

Course Structure

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. 

Detailed Course Content

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

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.

Course Planning

 SubjectsText References
1Richiami di modelli retti da equazioni alle derivate parziali: equazioni di Poisson, del calore e delle onde.
2Richiami di buona positura del problemi per le equazioni differenziali della Fisica Matematica.
3Equazione del calore. Richiami su alcuni procedimenti per ottenere soluzioni esatte in casi particolari: metodo di Fourier e di separazione delle variabili.
4Metodo di Eulero in avanti. Analisi della stabilità: metodo di von Neuman. Metodi impliciti: schema di Eulero all'indietro e di Crank-Nicholson.
5Sistemi tridiagonali.Equazioni del calore con coefficienti variabili.
6Consistenza, convergenza e stabilità dei metodi alle differenze finite per problemi ai valori iniziali. Teorema di equivalenza di Lax (enunciato).
7Equazione del calore in più dimensioni. Metodi a passi frazionari. Metodi Alternate Direction Implicit (ADI).
8Equazioni ellittiche. Richiami di teoria.
9Metodo alle differenze finite per l’equazione di Poisson su griglie Cartesiane. Discretizzazione di tipo vertex-center e cell-center.
10Il problema delle condizioni al contorno (condizioni di Dirichlet e di Neumann)
11Metodi di tipo level set e ghost point per il trattamento di geometrie arbitrarie.
12Medoto Multigrid per la risoluzione del relativo sistema algebrico sparso (cenni).
13Equazioni iperboliche. Singola equazione scalare lineare. Il metodo delle caratteristiche.
14Metodi alle differenze finite. I metodi a tre punti: upwind, Lax-Friedrichs e Lax-Wendroff
15Consistenza e stabilità. Condizione di Courant-Friedrichs-Lewy e dominio di dipendenza dai dati.
16Metodi del primo ordine e del secondo ordine. Equazione modificata, dissipazione e dispersione.
17Equazione di Burgers. Medoto delle caratteristiche. Soluzioni discontinue