The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge-Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.
In this Note, we present results of the numerical simulation of transition to turbulence for a purely oscillatory channel flow. These simulations were performed for various values of the Reynolds number, the so-called Stokes parameter being equal to 4. The methodology used for the flow simulation relies on a combination of finite element space approximations with timediscretization by operator splitting; it has shown to be very effective, even when it is applied to relatively complex domains with strong expansions at the inlet and outlet of the channel. The numerical results obtained agree qualitatively well with previous experiments by other investigators.
received his Ph.D. in mathematics from the University of Houston (USA), in 1996. He also was a postdoctoral fellow and assistant visiting professor of the same institution during the period 1999-2002. Since 2002 he is full professor of applied mathematics at the Universidad Autonoma Metropolitana-Iztapalapa (UAM-I) in Mexico City, and currently is a Fellow of the National Research System in Mexico, level 2. He has participated in numerous research projects and published several research articles, reports, conference proceedings and book chapters, mainly in CFD, numerical solution of PDE, mathematical modeling and computer simulation. He has been supervisor of several graduate students, and has participated in numerous academic committees. It has also helped organize numerous national and international meetings in Mexico, and he has participated in many others abroad. Professor Juárez was recently elected member of the Main Board of the Mexican Mathematical Society for the period 2012-2014.
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