Modelling and scientific programming

Objectives To learn to develop computer programs that are  Correct  Efficient  Adaptable  Portable

Remarks Price of computers  Cost of programming development  Clear programmes  time for development   costs  No unique solution  choose most appropriate solution

Content Course : 10h Introduction Errors and numerical stability Numerical analysis Linux Fortran 90 C++ Practical tutorials (20 hours) Project

Chapter 1 : Introduction Scientific research traditionally involves 2 main activities Experiments Modelling Observations Theories Predictions Suggest theories Test theories Interpret experiments Suggest theories confrontations

Classical mechanics ≡ Celestial mechanics  Horoscopes   observe sky  collect data  contradict Ptolemy’s geocentric model  introduce heliocentric model 85 – 165 AD Classical mechanics

 Nicolaus Copernicus heliocentric model circular trajectories

Develop observational tools and methods  Tycho Brahe Collect accurate data

 Johannes Kepler  Fit curves to data  3 laws Elliptical orbit for Mars

 Galileo Galilei modern scientific method  Observe : pendulum  Experiment : inclined plane free fall from the Tower of Pisa  Model Technical progress : optics  telescope  jump in the quality of observations

 Isaac Newton Develop mathematical tools (calculus) Unify apparently disconnected facts movements of planets movement of the Moon falling bodies  3 fundamental laws of classical mechanics  law of gravitation

Since Newton … Test theory  predict comets, new planets, … Improve methods  Corrections : 3-body problem, rigid body, tidal forces, precession, nutation, …  New technologies  accurate measurements  Many-body problems : Euler, Lagrange, Laplace, Gauss, …  complex methods to approximate motions of bodies = celestial mechanics  Transformation and integration of equations of motions = analytic mechanics Improve theory  Special relativity  General relativity Halley

Numerical approach = third approach, complementary to the 2 others Experiments ModellingNumerical simulation Suggest model Generate data Suggest theories Test theories Interpret experiments Suggest experiments Model real processes Suggest experiments Analyse data Provide equations Interpret results Suggest theories Allow accurate calculations large calculations

Advantages of numerical simulation Relatively easy to implement Economical Efficient Quantitative Applicable to complex systems Allow numerical experimentation Allow systematic investigation (scaling laws) Applicable to apparently very different systems, described by similar equations