CFU

9

Length

0 Weeks

Semester DD

First

* Fondamentali di numerica, errori di troncamento e arrotondamento

* Introduction to numerics, roundoff and truncation errors

* Ordinary differential equations (ODE)

- Stability

- Explicit and implicit methods

- Runge-Kutta e Dormand-Prince ODE5(4) and ODE8(7)

- Gear's Methods

- Applications: Strange attractors and caos, caotic pendulum and bifurcations,

Lyapunov exponents and fractality

* Introduction to molecular dynamics

- Development of an efficient simulator O(N)

- Applications: study of a Lennard-Jones liquid (phase transitions, diffusivity)

* Solutions of linear systems: direct and iterative methods (CG, GMRES)

* Partial differential equations (PDE)

- Elliptic, Parabolic and Iperbolic classifications

- Discretizations and stability (FDM, FVM, FEM).

- Numerical solutions of Poisson and Fourier Heat equations

- Numerical Solutions of Navier-Stokes equations

Basic numerics

oTruncation and round-off errors

Algoritmi numerici di base

oSearch of zeros (bisection, Newton)

oLinear solvers (direct: LU, iterative: CG, MINRES, GMRES)

Quadrature

oSimpson’s rules and Gaussian quadratures, Gauss-Lobatto

Equazioni differenziali ordinarie (ODE)

oStability

oExplicit Methods (Runge-Kutta 45, Dormant-Prince, PC, Adaptive)

oMetodi impliciti

Attractors e Caos

oAttrattore di Lorenz (derived from Benard cell model)

oLogistic Map, Bifurcations and Feigenbaum theory.

oLyapunov exponents.

Brief overview of Molecular Dynamics

oA Lennard-Jones gas (development of a code with periodic BC)

Spectral Methods

oFourier and Chebyshev methods (solve Burger’s equation).

Partial differential equations

oParabolic, Elliptic, Hyperbolic. Stability issues.

Numerical solutions of Navier-Stokes (Finite Volume Method)

Students acquire the ability to solve physical problems with the help of a computer as numerical instrument. During this lecture course non-trivial examples of numerical solutions to physical models are tackled, among which numerical solutions to Ordinary Differential Equations (ODE) and Partial differential equations (PDE). In particular problems related to discretizations and finite precision arithmetics are presented and solutions discussed.

ODE solvers are developed and applied to analyze strange attractors and caotic trajectories.

Methods to compute critical points, bifurcations, Feigenbaum numbers, Lyapunov exponents and fractal dimensions are presented and discussed.

Particularly oriented to theorist in phisics and statistical physics, a classical molecular dynamics is developed and applied to study the Lennad-Jones gas, modelling solid-liquid phase transition and computing radial distributions functions and self-diffiusion coefficients.

In the last part of the course, solutions to PDEs are tackled, either elliptic (e.g., Poisson equation), parabolic (e.g. heat equation) or hyperbolic (advection and Navier-Stokes).

In all cases discretization and stabilization methods are presented. In order to enable solutions of such equations, iterative methods such as Conjugate-Gradients and Krylov subspace methods are presented and implemented.