Prof. G. Niccolini
(S3, elective, 3 ECTS)
|During the mathematical modeling of a physical problem chances are that one will obtain differential equations, either Ordinary Differential Equation (ODE) or Partial Differential Equation (PDE). In astrophysics the need for numerical simulations is even more critical since the objects under study are far from being located in the lab. A large variety of physical problems are described in that way such as e.g. fluid dynamics, heat conduction, n-body simulation, radiative transfer, etc. Solving the equations of mathematical physics numerically is a good playground to face the classical problems of numerical analysis such as interpolating, integrating, solving linear systems of equations, studying the stability of numerical schemes etc … The student will acquire all the necessary theoretical background to solve differential equations numerically and will apply the freshly acquired material in small illustrative computer projects.
|Knowledge and Understanding:
|The student will learn how to solve numerical problems focused on the solution of differential equations, using various techniques from numerical analysis, such as the finite difference method. He/she will learn how to build and study the behavior of various numerical schemes used to solve the classical equation of mathematical physics such as for instance the heat equation, the wave equation etc …
|Applying Knowledge and Understanding:
|The student will develop computer programs (using C++ and python) to solve for classical problems of mathematical physics as a direct application of the new mathematical material introduced in this series of lectures in an hands-on approach.
|Mathematics knowledge : three-year university degree in mathematics and/or physics Physics knowledge : three-year university degree could be helpful but not mandatory.
|Part 1: Fundamentals • Round-off error and the floating point number system. • Truncation errors. • Interpolation • Linear systems of equations(a) Iterative methods for sparse systems (Jacobi, Gauss-Seidel, Successive Over-Relaxation, conjugate gradient, . . . ) • Finite Difference (FD) approximations of differential operators • Discrete Fourier Transform Part 2: Partial/Ordinary Differential Equations Solving ODE numerically • One-step methods (Euler, RK, Taylor series, explicit vs implicit methods) • Multi-step methods (Leap-Frog, Adams-Moulton, Adams-Bashforth)•Stability analysis : notion of zero and absolute stability, characteristic polynomial(a) Solving PDEs • Boundary conditions • Truncation error and consistency • Convergence and stability • Von Neumann stability criterion Part 3 : practical works Short introduction to programming. Development of computer programs to solve linear PDE of mathematical physics(wave equation, Schrödinger, diffusion, advection etc …)
|Description of how the course is conducted
|The course consists in a series of lectures, complemented by sessions of problem solving and practical works. Lecture notes in the form of a pdf document are provided, containing the material addressed in the course with a reservoir of exercises and a step by step description of numerical projects focused on the resolution of partial differential equations.
|Description of the didactic methods
|A thorough description of each topics is made in parallel to exercise sessions as a direct application to the presented new materials. The student will learn by intense training and personal work. He/she will also have to face actual problems in the form of small numerical/programming practical work where the interaction between student will be privileged . Students will finally work by pair on a larger project to complement their skills in an hands-on approach.
|Description of the evaluation methods
|Homeworks (10%), written exam (40%) and project (50%)
|• A First Course in Numerical Analysis, Ralston, A. and Rabinowitz, P., Dover Publications, Inc., 2001 • Finite difference methods for differential equations, LeVeque, R.J., 2007 • Numerical methods for engineers and scientists, Hoffman, Joe D and Frankel, Steven, CRC press, 2001 • Finite volume methods for hyperbolic problems, LeVeque, Randall J, Cambridge university press, 2002