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Academic Year 2023/2024 - Teacher: Giovanni RUSSO

Expected Learning Outcomes

The course provides an overview of some methods used in the numerical solution of systems of equations that describe the motion of fluids, both compressible to incompressible. Some general concepts (such as those relating to hyperbolic systems of laws of conservatione, and related numerical methods) can be used in a much broader context.

Course Structure

The course consists in lectures and exercise sessions, during which some of the method illustrated in class will be implemented on the computer. 

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.

Detailed Course Content

Elements of theory of hyperbolic systems. Wave propagation. Single scalar equation. Viscosity and entropy solutions. Hyperbolic systems: linear, semilinear and quasilinear. Riemann invariants. Jump conditions and entropy conditions. 

Euler equations of compressible gas dynamics. Deduction of the Euler equations. Rankine-Hugoniot conditions. Simple waves in gas dynamics. Polytropic gas. Isentropic gas dynamics. Riemann problem. Boundary conditions.

Numerical methods for conservation laws. Finite volume methods. Three point methods: upwind methods, Lax-Friedrichs method and method of Lax-Wendroff. Godunov method and its properties. The numerical flux function. of high-order construction methods. high-order reconstructions essentially non oscillatory (ENO). Weno reconstructions. Finite difference methods conservative. Integration over time: Runge-Kutta methods SSP (Strongly Preserving Stability). Treatment of source terms. Runge-Kutta methods IMEX (IMplici-Explicit) for the time integration.

Incompressible fluid dynamics. Deduction of the  incompressible Euler and Navier-Stokes. Finite difference methods for Euler and Navier-Stokes equations in primitive variables. Method of projections of Chorin and  MAC type (Marker and Cell)  discretization. Penalty methods for problems in domains with obstacle. Vorticity-stream function formulation for the Navier-Stokes equations.

Equations of shallow water. Deduction of the Saint-Venant model for the shallow water. Analogy with the isentropic gas dynamics. Finite volume methods and finite difference for the SV equations in one and two spatial dimensions.

Practise. The course includes exercises in which the main methods are implemented. In particular, they will be implemented and compared several methods for the solution of the compressible Euler equations and of the Navier-Stokes incompressible.

Remark. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus

Textbook Information

Textbooks on Computational Fluid Dynamics and related topics

The following are some books that deal several topics related to CFDm and that can be adopted during the course. 

  1. John D. Anderson    Jr., Computational Fluid Dynamics, the basics with applications, McGraw Series in Mechanical Engineering, 1995.
    A classic on CFD. Written by a professor in aeronautical engineering, it is very applied. Not particularly sophisticated from the mathematical point of view. A little out of date.
  2. Dimitris Drikakis, William Rider, High-Resolution Methods for Incompressible and  Low-Speed Flows, Springer, 2005.
    Rather updated, it gives a simple description of the mathematical formulation of the equations of gas dynamics.
  3. Joel H. Ferziger, Milovan Peric, Computarional Methods for Fluid Dynamics, Springer, 2002.
    Definitely numerical orientation, very detailed on numerical schemes, but rather lacking on modeling and mathematical aspects.
  4. Randall Le Veque- Finite Volume Methods for hyperbolic problems, Cambridge University Press, 2004.
    Specialized on finite volume schemes for hyperbolic systems of conservation laws.
  5. Randall Le Veque - Numerical methods for conservation laws, Lecture Notes in Mathematics, ETH Zürich, Birkhaeuser, Second edition, 1999.
    Excellent to give a mathematical treatment of systems of conservation laws and some of the recent numerical methods type shock-capturing for various systems of conservation laws.
  6. Roger Peyret, Thomas D. Taylor, Computational Methods for Fluid Flows, Springer-Verlag, 1983. Very synthetic, it treats mainly issues on incompressible fluid dynamics. Very advanced when it came out, now it is also quite dated.
    Testo sintetico, tratta prevalentemente temi di fluidodinamica incomprimibile. Molto avanzato quando è uscito, adesso è anch’esso piuttosto datato.
  7. Pieter Wesseling, Principles of Computational Fluid Dynamics, Springer Series in Computational Mathematics, 1991.
    Good introductory book. It contains much more material than can be addressed in the course. A selection is required.
  8. G.B.Whitham, Linear and nonlinear waves, John Wiley & Sons, 1974.
    Excellent book on the mathematical models that describe wave phenomena.

Course Planning

 SubjectsText References
1Propagazione ondosa. La singola equazione scalare.Caso lineare e non lineare. Metodo delle caratteristiche. G.B.Whitham, Linear and nonlinear waves, John Wiley & Sons, 1974.
2Soluzioni di viscosità e condizioni di entropia.Godlewski, Edwige, Raviart, Pierre-Arnaud, Numerical Approximation of Hyperbolic Systems of Conservation Laws , Springer, 1996.
3Sistemi iperbolici: lineari, semilineari e quasi-lineari.
4Soluzioni deboli e condizioni di salto.
5Condizioni di entropia.
6Deduzione delle equazioni di Eulero e di Navier Stokes.
7Vaie forme delle equazioni di Eulero.
8Onde semplici in gas dinamica.
9Gas politropici. Gas dinamica insentropica.
10Condizioni di Rankine-Hugoniot, shocks a discontinuità di contatto.
11Problema del pistone e problema di Riemann.
12Condizioni al contorno.
13Metodi ai volumi finiti. Medodi a tre punti: metodi upwind, metodo di Lax-Friedrichs e metodo di Lax-Wendroff .
14Metodo di Godunov e sue proprietà.
15La funzione di flusso numerica.
16Costruzione di metodi di alto ordine.
17Ricostruzioni di alto ordine essenzialmente non oscillatorie (ENO). Ricostruzioni WENO.
18Metodi alle differenze finite di tipo conservativo.
19Integrazione nel tempo: metodi Runge-Kutta SSP (Strongly Stability Preserving).
20Trattamento dei termini di sorgente. Metodi Runge-Kutta IMEX (IMplici-EXplicit) per l’integrazione temporale.
21Deduzione delle equazioni di Eulero e Navier-Stokes incomprimibili.
22Metodi alle differenze finite per equazioni di Eulero e Navier-Stokes in variabili primitive.
23Metodo delle proiezioni di Chorin e discretizzazione di tipo MAC (Marker and cell).
24Metodi di penalizzazione per problemi in domini con ostacolo.
25Formulazione vorticity-stream function per le equazioni di Navier-Stokes.
26Deduzione del modello di Saint-Venant per le acque poco profonde. Analogia con la gas dinamica isentropica.
27Metodi ai volumi finiti ed alle differenze finite per le equazioni di SV in una e due dimensioni spaziali.
28Metodi well-balanced.