Numerical Schemes for Compressible Fluid Mechanics

By offering a complementary approach to experiment and modeling, numerical simulation has become one of the three pillars of scientific research. Fluid mechanics is one of the pioneering sectors in the use of this triptych, and obtaining well-adapted numerical schemes for fluid mechanics is a subject that has occupied numerical engineers for over 60 years. One of the difficulties lies in reconciling accuracy and robustness with reasonable computational costs, and the challenges can be quite different depending on the targeted applications. So, despite a great deal of work and advances in a subject that is still very topical, it's quite natural that there is no uniformly effective technique for all regimes.

The aim of this article, therefore, is not to propose a perfect scheme that doesn't exist, nor to detail all the very many techniques in an abundant literature, but rather to describe reasonably simple choices that are good candidates for getting started. There will necessarily be a certain subjectivity in these choices, but some alternatives and the most common avenues for improvement will be mentioned. The reader will find an entry point to a realistically programmable solution, capable of simulating compressible flows in standard configurations. They are also encouraged to program these methods themselves for unstructured 2D meshes. Since part of their effectiveness lies in the speed of calculations, it is important to test them to see their performance, their difficulties and how they can be optimized. Since this depends in part on the programming language, the machines that will perform the calculations and the programming style, there's no substitute for experience.

Similarly, readers are encouraged to delve deeper into the subject, depending on the applications that interest them. A large body of literature is available to provide solutions to most practical problems. Research is still very active on the subject, and new techniques are continually being developed to push back the limits of numerical simulation. The bibliography presented here is just a tiny fraction of the work in the field.

The methodology adopted in this document will be as follows. In each section, we will present the basic elements needed to properly introduce the concepts. These will sometimes be supplemented by remarks, or even small information points called "Pour aller plus loin" (to go further). As their name implies, they deal with complementary notions that are not necessary for general understanding. They are, however, important to know in absolute terms.

First, we'll look at the approximation of the Euler equations, then we'll look at techniques for taking viscosity/diffusion terms into account, and finally we'll say a few words...