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.
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.
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
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.