Lecturers: Enrique ZUAZUA and Aurora MARICA, Ikerbasque and BCAM Title: TOPICS ON NUMERICS FOR WAVE PROPAGATION Date and time: Mon, June 11, 2012 to Fri, June 15, 2012, 9:00 to 11:00 Program: 1.- Waves, history and applications 2.- Background on Fourier analysis. 3.- Finite difference discretizations. 4.- Group velocity and concentrated waves. 5.- Dispersive properties. 6.- Control of vibrations. 7.- Heterogeneous media Contents: In this series of lectures we will make a self-contained introductory presentation of the theory of wave propagation, their numerical approximation and computer implementation and some of the modern applications. We will start by making a brief historical presentation of some of the main applications that led to the development of the mathematical theory of wave propagation. We will then present the basic material of Fourier analysis needed to analyze this phenomena in detail. We shall also discuss in detail finite difference numerical approximation schemes and present the basic ingredients of the theory of stability an convergence. We will later discuss in detail the notion of group velocity and also the existence and possible constructions of concentrated waves. We will continue analyzing some of the main dispersive phenomena arising when performing numerical approximations. An application to control the control of vibration structures will also be presented. We will end up with a brief discussion of the main features arising when extending the theory to heterogeneous media and non uniform meshes. Methodology. This course is oriented to Ph students with basic background in Differential Equations and Numerical Analysis and interest in Partial Differential Equations, Numerical Analysis, Control Theory and/or its applications. The course will avoid major technical difficulties and will be accessible to a large audience. The material will be presented in five two hour lectures that will be complemented by Computing Laboratory exercises that will be developed jointly with students in the MatLab and FreeFem environment.

 

 

References: S. Ervedoza and E. Zuazua, The Wave Equation: Control and Numerics, in ``Control and stabilization of PDE's", P. M. Cannarsa and J. M. Coron, eds., “Lecture Notes in Mathematics”, CIME Subseries, Springer Verlag, to appear (www.bcamath.org/zuazua) Hecht F., FreeFEM++, Third Edition, Version 3.12, http://www.freefem.org/ff++ Kalechman M. (2009), Practical Matlab. Basics for engineers, CRC Press, Taylor and Francis Group. LeVeque, R. J. (1992). Numerical methods for conservation laws. Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.} Quarteroni A. and Valli, A. (1998). Numerical approximation of Partial differential Equations, Springer, Springer Series in Computacional Mathematics, 23. Sethian, J. A. (2002), Level set methods and fast marching methods}. Cambridge Monographs in Applied and Computational Mathematics, Cambridge University Press. Strikwerda J. (2004), Finite difference schemes and partial differential equations, SIAM. Trefethen L.N. (1982), Group velocity in finite difference schemes, SIAM Review 24, 113-136. Trefethen L.N. (2000), Spectral methods in Matlab, SIAM. Vichnevetsky, R. and Bowles, J.B. (1982). Fourier analysis of numerical approximations of hyperbolic equations.} SIAM Studies in Applied Mathematics, {\bf 5}, SIAM, Philadelphia, Pa. Zuazua, E. (2003). Introducción al Análisis Numérico de Ecuaciones en Derivadas Parciales de Evolución. (http://www.uam.es/enrique.zuazua) Zuazua, E. (2004). Métodos numéricos de resolución de Ecuaciones en Deivadas Parciales. (www.bcamath.org/zuazua)