Title: Simulating fluids with a computer: Introduction and recent advances Show Chinese title

Title: 用计算机模拟流体：引言与近期进展

Abstract: In this article, I present recent methods for the numerical simulation of fluid dynamics and the associated computational algorithms. The goal of this article is to explain how to model an incompressible fluid, and how to write a computer program that simulates it. I will start from Newton laws "$F = ma$" applied to a bunch of particles, then show how Euler's equation can be deduced from them by "taking a step backward" and seeing the fluid as a continuum. Then I will show how to make a computer program. Incompressibility is one of the main difficulties to write a computer program that simulates a fluid. I will explain how recent advances in computational mathematics result in a computer object that can be used to represent a fluid and that naturally satisfies the incompressibility constraint. Equipped with this representation, the algorithm that simulates the fluid becomes extremely simple, and has been proved to converge to the solution of the equation (by Gallouet and Merigot). Show Chinese abstract

Abstract: 本文中，我介绍了用于流体动力学数值模拟的近期方法以及相关的计算算法。 本文的目标是解释如何建模不可压缩流体，以及如何编写一个可以模拟它的计算机程序。 我将从应用于一组粒子的牛顿定律“$F = ma$”开始，然后展示如何通过“退后一步”并将流体视为连续体来从中推导出欧拉方程。 接着 我将展示如何编写一个计算机程序。 不可压缩性是编写一个模拟流体的计算机程序的主要困难之一。 我将解释计算数学领域的最新进展如何产生一个可以用来表示流体的计算机对象，并且自然地满足不可压缩性约束。 配备了这种表示方法，模拟流体的算法变得极其简单，并且已经被证明收敛于方程的解（由Gallouet和Merigot证明）。