Course program
Motivations and applications of numerical analysis to the study of ordinary differential equations (ODE), dynamical systems, mechanical systems, vector fields, geometric interpretation. The Cauchy problem for ODEs, differential and integral formulation, existence, local and global uniqueness of solutions. Lyapunov stability. Asymptotic stability. Gronwall's lemma, sufficient conditions for the stability of ODE. One-step numerical methods for the solution of ODEs, general ideas on the discretization of the differential and integral formulation. Explicit and implicit methods, first examples (forward Euler, backward Euler, Crank-Nicolson, Heun). Analysis of one-step methods, increment function, local truncation error, consistency and order of consistency with examples. Zero-stability of one-step methods for ODEs. Discrete Gronwall lemma, sufficient conditions for zero-stability. Convergence and convergence order. Lax-Richtmyer equivalence theorem.
Explicit convergence analysis in the case of the forward Euler method. Stability estimates in the presence of rounding errors. Absolute stability of one-step methods for ODE, linear model problem. Region of absolute stability, explicit calculation for forward Euler, backward Euler, Crank-Nicolson, Heun methods. Runge-Kutta methods, derivation, Butcher table, sufficient conditions for consistency, zero-stability. Construction of explicit RK methods, order of convergence in relation to the number of stages, absolute stability, characterization of the stability function.
ODE systems: linear case, resolution by implicit methods; nonlinear case, resolution by fixed point iterations and Newton's method. Application to the harmonic oscillator in one dimension.
Linear Multi-step methods (LMM), general form, explicit and implicit methods, trigger values, consistency and order of consistency, first examples (Midpoint, Simpson). Parallelism between linear differential equations with constant coefficients and difference equations. Algebraic conditions equivalent to consistency for an LMM. First and second characteristic polynomial for a LMM, roots condition, equivalence with zero-stability. Characteristic polynomial for a LMM applied to the linear model problem, absolute roots condition, equivalence with absolute stability. Qualitative behavior of the numerical solutions of a LMM for a homogeneous equation, comparison between the zero-stability and the roots condition, comparison between the absolute stability and the absolute roots condition. Equivalence theorem for LMM, first and second Dahlquist barrier. Adams methods explicit (Adams-Bashforth (AB)) and implicit (Adams-Moulton (AM)). Construction of AB and AM methods using interpolating polynomials, consistency, zero-stability and absolute stability. The boundary locus, a "trick" for calculating the boundary of the region of absolute stability. BDF (Backward Differentiation Formula) methods, construction by interpolating polynomial, consistency, zero-stability and absolute stability. Predictor-Corrector methods, PEC and P(EC)^m examples. Order of methods P(EC)^m. Introduction to variable step methods, a posteriori estimate of the truncation error for step adaptivity.
Prerequisites
Differential and integral calculus, elements of ordinary differential equation theory (existence and uniqueness, concepts of stability, analytical resolution of linear equations), linear algebra, C or Matlab programming. The Numerical Analysis course is strongly recommended.
Books
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer.
Lecture notes in pdf and Matlab scripts examples.
Frequency
Attendance is optional but strongly recommended given the way it is carried out.
Exam mode
The exam consists of a theory test and a practical test. The theory test takes place in the classroom (max 3 hours) and includes a written test with questions/exercises relating to the theoretical aspects covered in the course. The practical test takes place in the laboratory (max 3 hours) and involves the creation of a code for the numerical resolution of an ODE system, using one of the methods studied during the course, as well as the graphical display of the results. The final grade will take into account, equally, the results obtained in both tests.
Bibliography
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer.
Lesson mode
The course takes place entirely in the laboratory in order to use the available computers not only for the exercise and code development part, but also as an aid to understanding the theoretical concepts covered.
