This module aims to teach students high-precision numerical methods associated with numerical resolution in fluid mechanics. Following the modules on conventional numerical methods (finite differences, finite volumes, finite elements, matrix inversion, etc.), it covers high-order spatial discretization methods (compact finite differences, spectral methods), temporal integration of high-order PDEs, implicit and semi-implicit methods, as well as Krylov methods for solving large linear systems (conjugate gradient, Biconjugate, GMRES). These methods are finally applied to a concrete case in fluid mechanics (Korteweg-de Vries equation, nonlinear Schrödinger equation, Kuramoto-Sivashinsky equation, etc.).