The course aims to introduce the simulation of systems modelled by differential equations or in terms of deterministic or stochastic processes.
The student will acquire a general knowledge of the most common numerical methods for simulating complex systems in the time domain and more generally, by sampling the configuration space.
The course will promote in the students the following specific learning objectives:

• to be able to implement in a program some of the classical simulation techniques;
• to be able to estimate precision and accuracy of a numerical simulation;
• to be able to identify the main causes of numerical instability in a simulation;
• to be able to estimate the error dependency on the simulation parameters;
• to be able to estimate the convergence order of a specific simulation method;
• to be able to compare the results of simulations obtained with different methods;
• to acquire a basic knowledge of some modern simulation techniques;
• to be able to choose the simulation technique most appropriate for a specific mathematical model;
• to be able to discuss the simulation results by presenting formulas, code and graphics;
• to learn the basics of Python programming and the use of some of the main Python libraries for numerical and symbolic computation and for graphics output

1. Course introduction
   1.1 The Python language: a crash introduction

2. Ordinary differential equations
   2.1 The Euler method
   2.2 Orders of convergence
   2.3 Higher order schemes, and Runge-Kutta methods

3. Partial differential equations: convective problems
   3.1 The convective motion in 1D
   3.2 Numerical stability and CFL condition
   3.3 The diffusion equation in 1D
   3.4 Convection and diffusion together: the Burgers equation

4. Partial differential equations: diffusive problems
   4.1 Heat equation in 2D: explicit methods
   4.2 Reaction-diffusion equations

5 Stochastic systems
   5.1 Stochastic processes, random walks, Ornstein-Uhlenbeck model
   5.2 Distribution, Metropolis-Hastings
   5.3 Ising model as an application of MH

6. Partial differential equations: elliptic problems
   6.1 Laplace equation, and Jacobi method
   6.2 Poisson equation
   6.3 Gauss-Seidel
   6.4 Example: a ball on an elastic drum

7 Simulations based on emerging dynamics
   7.1 Cellular automata
   7.2 Lattice gas cellular automata

It is crucial to have already attended a courses on Logic, Geometry and Algebra and on Calculus since derivatives, integrals and matrix calculs are relevant for all topics of the Numerical Simulation course.

Having attended a course on Probability and Staitistics is very useful in order to be able to understand stochastic simulations.

The basics of programming are very useful.

Knowledge and understanding: the student will learn the main simulation methods in the time domain, including the Euler method and its higher-order generalisations, as the Leap-Frog and Runge-Kutta methods. The student will be able to formulate and solve numerically ordinary differential equations and some of the most common partial differential equations. The student will be able to simulate stochastic systems, also with several degrees of freedom. 
Applying knowledge and understanding: the student will be able to choose the simulation method most appropriate for a specific mathematical problem, and will be able to transpose it into an algorithm. The student will be able to implement a simulation algorithm that uses assigned data, generates and presents in graphics form numerical results.
Making judgements: the student will be capable of evaluating the accuracy of simulation results and of estimating the dependency of the error on the choice of the simulation parameters, for instance, by identifying the order of convergence of an algorithm for the numerical integration of a differential equation.
Communication skills: the student will be able to present the results of his/her simulation work. In particular, he/she will be able to write down a structured, well-documented code, and to present the simulation results in graphics form in order to communicate correctly and effectively.
Learning skills: the student will be in a position to study in greater depth specific topics in numerical simulation not discussed during the course by making use of scientific books and specialised publications in the field.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Frontal lessons

Attendance

Attending the course is not an obligation, but it is strongly recommended.

Course books

The study material and the relevant bibliography will be provided by means of the Moodle platform.

Assessment

The student will be tasked to study a topic of her/his interest concerning numerical simulation and present it in a seminar.
The content of the seminar will be discussed, possibly reviewing the student's knowledge of other topics in the course.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

The same as for attending students, thanks to the availability of course material through the Moodle platform.

Course books

The same

Assessment

The same

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.