69721 - Laboratory of Numerical Geophysics

Course contents

Module 1 – Theory

Ordinary differential equations (ODEs). Analytical solutions, linear, autonomous, homogeneous, nonlinear equations, Theorem of existence and uniqueness. Discretization of ODE, explicit and implicit methods, theta-methods, local and global accuracy. Adams-Bashforth and Adams-Moulton methods. Runge-Kutta methods.

Partial differential equations (PDEs). Discrete differential operators. Transport equation: analytical solution and discretization, method of characteristics, stability analysis, Von Neumann criterion and CFL criterion, accuracy of explicit and implicit numerical methods. Heat diffusion equation: analytical solution and discretization, Von Neumann stability, accuracy of explicit and implicit methods. Wave equation: analytical solution and discretization, stability according to Von Neumann and CFL, accuracy, boundary conditions.

Module 2 – Laboratory

Design and implementation of numerical algorithms on exercises provided by the teacher concerning ODE (Adams and Runge-Kutta methods) and PDE (transport equation, heat equation, wave equation).

The student will develop the numerical codes on his own PC (or any computer made available in the laboratory) and will verify the stability and accuracy conditions through comparisons with the respective analytical solutions. This activity will be carried out with the MatLab software.

Readings/Bibliography

Module 1 – Theory

Slides presented during the lectures.

Module 2 – Laboratory

Outlines of the exercises performed in the laboratory

Teaching methods

Module 1 – Theory

Classroom lectures, projection of slides.

Module 2 – Laboratory

Laboratory exercises carried out on own computer.

Attendance of this training activity requires the prior participation of all students to modules 1 and 2 of training on safety in the places of study, through e-learning.

Assessment methods

The final test is joint for the two modules, exclusively oral, with duration ranging between an hour and an hour and a half. It will concern the theory as well as the numerical tests carried out in the laboratory. The examinee will be asked to show, discuss and run the numerical codes solving the differential equations developed in the laboratory.

Students with SLD or temporary or permanent disabilities: it is recommended to contact the responsible University office in time (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). It will be its responsibility to propose any adaptations to the student concerned, which must in any case be submitted, 15 days in advance, to the approval of the teacher, who will also assess the opportunity in relation to the educational objectives of the course.

Teaching tools

Projector; PC and MATLAB.

Office hours

See the website of Filippo Zaniboni