This book aims at covering the foundations of high accuracy computing methods within the framework of Computational Fluid Dynamics (CFD) in an era of rapidly developing and evolving hardware and software.

From the hardware point of view, huge parallel machines with tens of thousands cores are installed at national facilities and research laboratories giving the practioners of scientific computing tools that they could not have dreamt of a decade ago. The advent of Graphical Processing Units (GPUs) also modifies the course of CFD as everyone tries to strain the computational tools to their last bits and extracts the highest speed-up. This is not surprising as one of the unsolved problems in classical physics is the understanding and control of turbulence in nature and technological applications.

From the software viewpoint, the advent of commercial packages including mesh generators, solvers and graphics tools, provide the numericists with appealing users interfaces and deliver numerical results for extremely different and various problems involving complicated geometries, peculiar boundary conditions and complex physics to be captured. This has had a major impact on the CFD community.

A question that is often raised consists in asking “Why should we not use the simplest schemes and run them on millions (billions) of processors?” The problem as we will discover rapidly is that simple schemes are very often too naive and lead to numerical disaster. We cannot assume that our intellectual indolence will be compensated by the computer’s power. At the end of the day, a bad method will produce inconsistent and poor results.