1.1.1 Motivation

Numerical methods are a part of the problem solving skills that are expected to be mastered by most of the university graduates working in a quantitative field. The same fundamental concepts of Brownian motion, convection, diffusion, dispersion and non-linearity are used to simulate applications in telecommunication (collisions of data-packets in a network, solitons in optical fibers), economics (stock options), biology (transport in cellular tissues), engineering (heat transfer, pollution) and social sciences (behavior of people in a crowd). Quantitative answers for the real world can generally be obtained only from computations.

The goal of this course is to cover a wide range of numerical methods with simple examples to learn the building blocs that are often used in complex simulation codes. Indeed, a broader knowledge is often decisive to choose the method that is best suited when developing a new code: in this course, we will attribute an equal importance to finite-differences, finite-elements, fast Fourier transformation, Monte-Carlo simulations and Lagrangian schemes. In each case, the algorithm will be illustrated with a variety of prototype problems including the Fokker-Plank (transport), Black-Scholes (stock options), Burger (shock waves), Korteweg-DeVries (solitons) and the Schrödinger equations (quantum mechanics).

A strong emphasis is put on a problem based learning, where all the steps are discussed starting from the derivation, implementation and execution of a model to the discussion of the merits and limitations that appear when different methods are compared in the same setting.

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