My collaborator Changhui Tan and myself just submitted a new research paper, entitled An Exact Rescaling Velocity Method for some Kinetic Flocking Models.
In this work, we discuss kinetic descriptions of flocking models, of the so-called Cucker-Smale and Motsch-Tadmor types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potential. We introduce a new exact rescaling velocity method, inspired by a recent work of Filbet and Rey, allowing to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.
I am co-organizing the 2013 KI-Net Young Researcher Workshop, Kinetic and macroscopic models for complex systems, which will take place at the Center for Scientific Computing and Mathematical Modelling (CSCAMM) from October 14 to October 18, 2013. Some spots are still available to apply. Here is the abstract of this meeting:
Complex systems have emerged as a dominant topic in modern science. From animal behaviors to social networks, the impact of complex systems are noticeable on a daily basis. New mathematical models have been developed to apprehend those phenomena. For instance, kinetic models constitute a valuable resource to model complex systems with detailed interactions. On the other hand, macroscopic models focus on the evolution of average quantities which offer a better understanding of systems at larger scales. Both approaches provide new mathematical difficulty (e.g. pattern formation, stability analysis, numerical validation) and ultimately challenge our understanding of complex systems.