99560 - FINITE DIFFERENCE METHODS FOR DIFFERENTIAL EQUATIONS M

Learning outcomes

The students learn and are able to implement the finite difference method for the simulation of linear and nonlinear acoustic systems. These include lumped systems such as the harmonic oscillator, the Duffing oscillator, the van der Pol oscillator, oscillators with nonlinear damping, as well as distributed vibrating systems such as cables, strings, bars, membranes and plates, in both linear and nonlinear regimes. Furthermore, the students learn the mathematical methods used in the analysis of such systems, such as Fourier and Laplace methods for linear systems, and pertubation methods for nonlinear systems.

Course contents

1. Finite difference intro. Approximations to temporal operators. Truncation error. Laplace and z transforms and their use in the analysis of continuous and discrete linear, time-invariant systems.

2. Harmonic Oscillator. Finite difference schemes. Stability. Energy methods. Modified equation methods. Exact discretisation. Losses. External forcing. Solution via Laplace transform. Green's function and convolution theorem.

3. Nonlinear oscillators. Phase plane and equilibrium points. Cubic oscillator (Duffing). Analytic methods used to obtain the amplitude-frequency relationship (Lindstedt-Poincaré, multiple scales). Lossy oscillators: dry friction, van der Pol. Limit cycles. Finite difference schemes. Energy methods. Explicit, linearly-implicit and fully-implicit schemes.

4. Banks of oscillators. Eigenvalue analysis for linear systems. Transition to continuous systems.

5. Spatial discretisations. Finite differences, Galerkin-type methods, finite elements.

6. 1D wave equation. Phase and group velocities. Fourier solution: modes. Losses and forcing. Boundary condition of Neuman and Dirichelet type. Boundary conditions with finite impedance. Finite difference solutions. Courant–Friedrichs–Lewy (CFL) condition: stability cone and numerical dispersion.

7. Sparse linear systems. Tridiagonal systems. Thomas method and Jacobi method. Gauss-Siedel method.

8. Bars. Longitudinal (hyperbolic) and transverse (parabolic) waves. General elastic boundaries.

9. 2 dimensional finite difference operators. 2D wave equation. Polar and cartesian grids.

The course contents may vary according to the cohort's needs. The course will be taught in English.

Readings/Bibliography

Class hand-outs by the lecturer. Other useful texts:

- On finite differences

• R.J. LeVeque, Finite Difference Methods for Ordinary and Partia lDifferential Equations. Steady State and Time Dependent Problems. SIAM, Philadelphia, USA, 2007.
• J. Strikwerda, Finite Difference Schemes and Partial Differential Equations. SIAM, Philadelphia, USA, 2004.
• S. Bilbao, Numerical Sound Synthesis. Wiley, Chichester, UK, 2009.

- On finite elements

• T.J.R. Hughes, The Finite Element Method. Dover, Minerola, USA, 2000.

- On dynamical systems

• L. Meirovitch, Fundamentals of Vibrations. Waveland, Long Grove, USA, 2001.
• A. H. Nayfeh, Professor D. T. Mook, Nonlinear Oscillations. Wiley, Weinheim, Germany, 2004.

Teaching methods

Class (3hrs/week)

Matlab tutorials (2hrs/week). During the tutorials, the students will implement the numerical methods seen during class.

Assessment methods

The exam is project-based. Each student will work independently and submit a Matlab project, to be discussed with the lecturer during an oral exam. During the same exam, the students may be asked to answer more questions related to various other topics of the course.

Teaching tools

Class hand-outs. Matlab demos. Powerpoint presentations. Accelerometric measurement demo.

Since the course involves computer lab sessions, the students must take modules 1 and 2 on health and safety in the workplace, available at [https://elearning-sicurezza.unibo.it/]

Office hours

See the website of Michele Ducceschi