Course Details for A.Y. 2018/2019

Course Objectives

This is a monographic course on the numerical approximation of differential equations which is intended to provide both basic aspects on the topic, as well as more advanced ones. Together with the theoretical inspection on the numerical methods and their features, the course is also intended to provide the basic tools to develop a reliable software implementing the studied methods and analyze the obtained results on a selection of significant test problems.

Course Content

• The problem: Hadamard well-posed differential problems. Picard-Lindelof iterations. Existence and uniqueness of solutions. Continuous dependence on initial data and vector fields. Gronwall lemma. Examples of differential problems of interest in real applications.
• Basics on discretization of differential problems: difference equations, consistency, stability, convergence.
• Runge-Kutta methods. Formulation of methods. Butcher theory of order conditions, rooted trees and B-series. Error estimations, Richardson extrapolation, embedded methods. Stepsize control. Collocation methods. Implicit Runge-Kutta methods. Stability analysis.
• Linear multistep methods. Adams-Bashforth methods. Analysis of convergence, Lax theorem. Variable stepsize, Nordsieck formulation. Analysis of stability, boundary locus. BDF methods.
• Stiff problems. A-stability, L-stability, stiffly-stable methods. Schur polynomials, Routh-Hourwitz criterion.
• Structure-preserving approximation: the case of Hamiltonian problems. Basics on Hamiltonian dynamics, invariant preservation. Symplectic numerical methods. The non-symplecticity of linear multistep methods. Long-term properties of symplectic integrators. Quasi-conservation of invariants. Dynamics of multistep methods. Underlying one-step method. Backward error analysis.
• Numerical approximation of stochastic differential equations. Strong and weak solutions. Euler-Maruyama method, Milstein method. Mean-square convergence. Linear multistep stochastic methods. Stochastic Runge-Kutta methods. Mean-square and asymptotic stability.
• Oscillatory differential equations. Modulated Fourier series expansions. Long-term near conservation of total and oscillatory energy.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

Basic Numerical Analysis differential equations.

Assessment Methods and Criteria

Oral examination and a final project consisting in applying the developed software on a selection of test problems. The final project will be discussed in the same day of the oral examination.

Textbooks

• E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations , Springer. 2006.    
• E. Hairer, S.P. Norsett, G. Wanner, Solving ordinary differential equations I. Nonstiff problems , Springer. 1993.    
• E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems , Springer. 1996.    

Course page updates

• A.Y. 2017/2018
• A.Y. 2018/2019