401-2654-00L  Numerical Analysis II

 Semester Spring Semester 2020 Lecturers H. Ammari Periodicity yearly recurring course Language of instruction English

 Abstract The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation. Objective The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments. Content Chapter 1. Some basics1.1. What is a differential equation?1.2. Some methods of resolution1.3. Important examples of ODEsChapter 2. Existence, uniqueness, and regularity in the Lipschitz case2.1. Banach fixed point theorem2.2. Gronwall’s lemma2.3. Cauchy-Lipschitz theorem2.4. Stability2.5. RegularityChapter 3. Linear systems3.1. Exponential of a matrix3.2. Linear systems with constant coefficients3.3. Linear system with non-constant real coefficients3.4. Second order linear equations3.5. Linearization and stability for autonomous systems3.6 Periodic Linear SystemsChapter 4. Numerical solution of ordinary differential equations4.1. Introduction4.2. The general explicit one-step method4.3. Example of linear systems4.4. Runge-Kutta methods4.5. Multi-step methods4.6. Stiff equations and systems4.7. Perturbation theories for differential equationsChapter 5. Geometrical numerical integration methods for differential equation5.1. Introduction5.2. Structure preserving methods for Hamiltonian systems5.3. Runge-Kutta methods5.4. Long-time behaviour of numerical solutionsChapter 6. Finite difference methods6.1. Introduction6.2. Numerical algorithms for the heat equation6.3. Numerical algorithms for the wave equation6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimensionChapter 7. Stochastic differential equations7.1. Introduction7.2. Langevin equation7.3. Ornstein-Uhlenbeck equation7.4. Existence and uniqueness of solutions in dimension one7.5. Numerical solution of stochastic differential equations Lecture notes Lecture notes including supplements will be provided electronically.Please find the lecture homepage here:https://www.sam.math.ethz.ch/~grsam/SS20/NAII/All assignments and some previous exam problems will be available for download on lecture homepage. Literature Note: Extra reading is not considered important for understanding thecourse subjects.Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994.Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996.Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002.L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009.Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993.Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972.Walter: Ordinary differential equations, Springer-Verlag, New York, 1998. Prerequisites / Notice Homework problems involve Python implementation of numerical algorithms.