Distinguished Lecture Series in Mathematics 系列数学前沿学术讲座

报告题目：Conservative Semi-Lagrangian methods for Kinetic Equations

报告人：Prof. Giovanni Russo (Department of Mathematics and Computer Science, University of Catania, 95125 Catania, Italy)

报告时间：2021年11月25日（周四）16:00-17:30 PM (Beijing time), 9:00-10:30 AM (Rome Time)

报告地点:教二楼913

ZOOM URL: https://zoom.us/j/97412413730

Meeting ID: 974 1241 3730 Passcode: 321050

摘要：

In this talk, we overview a class of high order conservative semi-Lagrangian schemes for kinetic equations [1-2]. The schemes are constructed by coupling the conservative non-oscillatory reconstruction technique [1] with a conservative treatment of the collision term, obtained by either a discrete Maxwellian [3] or by an L^2-minimization technique [4]. Due to the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition, while the implicit treatment of the relaxation term based on L-stable Runge-Kutta or BDF time discretization enables us to avoid the stiffness problem coming from a small Knudsen number. Because of L-stability and exact conservation, the resulting scheme is asymptotic preserving for the underlying fluid dynamic limit. Several test cases confirm the accuracy and robustness of the methods, and the AP property of the schemes.

The method has been extended to the treatment of inert gas mixtures, and applied to compare different models in various regimes [5]. In particular, BBGSP model [6] allows the possibility of different scaling in the relaxation time for temperature and momentum, according to whether collisions are among molecules of the same species of molecules of different species. In some appropriate limit, this may lead to multiple temperature, multiple velocity Euler equations, which allow a more detailed description compared to the single-velocity single-temperature Euler equations derived by other models.

In general, such approaches use fixed velocity grids, and one must secure a sufficient number of grid points in phase space to resolve the structure of the distribution function. When dealing with high Mach number problems, where large variation of mean velocity and temperature are present in the domain under consideration, the computational cost and memory allocation requirements become prohibitively large. Local velocity grid methods have been developed to overcome such difficulty in the context of Eulerian based schemes [7-8]. In this talk, we introduce a velocity adaption technique for the semi-Lagrangian scheme applied to the BGK model. The velocity grids will be set locally in time and space. We apply a weighted minimization approach to impose global conservation, generalizing the L^2-minimization technique introduced by I.Gamba et al. We demonstrate the efficiency of the proposed scheme in several numerical examples.

An additional application of conservative SL schemes concerns the numerical simulation of Vlasov-type equations [9]. Here conservation of the scheme will provide some advantage over standard non conservative schemes for long time computation.

The research is conducted with the following collaborators: S. Boscarino, S. Y. Cho, M. Groppi, S.-B. Yun, JM Qiu, and T. Xiong

报告人简介:

Giovanni Russo obtained his master degree in Nuclear Engineering Magna cum Laude at the Polytechnic University of Milano, Italy, in 1982. He obtained a PhD in Physics at the University of Catania in 1987, under the guidance of Prof. Marcello Anile. He spent three years as a Visiting Research Scientist at the Courant Institute in New York. In 1990 he became Assistant Professor in Mathematical Physics at the University of L’Aquila, Italy, where he became associate professor in Numerical Analysis in 1992. Since 2000 he is full professor of Numerical Analysis at the University of Catania, Sicily, Italy.

His main research interests cover numerical methods for PDE’s, with particular attention to hyperbolic and kinetic problems, multi-scale methods, and various problems related to computational fluid dynamics. His current interest includes asymptotic preserving schemes, ghost fluid methods and multi-scale modelling of sorption kinetics.

He has more than one hundred scientific publications in international journals on various topics in applied and computational mathematics, and some books.

He has been coordinator of various PhD programs in mathematical subjects for more than 16 years, has been responsible of various national and international research projects, and has been visiting professor of several international institutions.

He acts as a member of the Editorial Board of the several journals including: SIAM Journal of Mathematical Analysis, Communication of Mathematical Sciences, Journal of Mathematics in Industry.

REFERENCES:

[1] S. Y. Cho, S. Boscarino, G. Russo, S.-B. Yun, Journal of Computational Physics, 432, pp. 110159 (2021).

[2] S. Y. Cho, S. Boscarino, G. Russo, S.-B. Yun, Journal of Computational Physics, 436, pp. 110281 (2021).

[3] L. Mieussens, Journal of Computational Physics, 162, pp. 429-466 (2000).

[4] I. Gamba, S. H. Tharkabhushaman, Journal of Computational Physics, 228, pp. 2012-2036 (2009).

[5] S. Boscarino, S. Y. Cho, M. Groppi, and G. Russo. “BGK models for inert mixtures: comparison and applications”, arXiv:2102.12757 (submitted), 2021.

[6] A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga, I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377.

[7] S. Brull, L. Mieussens, Journal of Computational Physics, 266, pp. 22-46 (2014).

[8] F. Bernard, A.Iollo, G. Puppo, Communications in Computational Physics, 16, pp. 956-982 (2014).

[9] T Xiong, G Russo, JM Qiu, Journal of Scientific Computing 79 (2), 1241-1270.

举办单位：首都师范大学数学科学学院、交叉科学研究院

北京国家应用数学中心、北京成像理论与技术高精尖创新中心

光场成像与数字几何北京市重点实验室

联系人：杨佳