Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations. Mathematics Subject Classification (2010): 82B40, 82C05, 35Q20, 65R10. Introduction by the Organizers The mini-workshop Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations was devoted to the numerical novel aspects of kinetic modelling. Kinetic modelling has become one of the most powerful tools in applied mathematics to bridge microscopic and macroscopic descriptions of many body systems during the last 30 years, see [12] for a review. The flexibility of the methods stemming from kinetic and large deviation theories have had numerous applications in physical, biological and technological problems. They typically involve a huge number of individuals, showing some sort of “collective behaviour”, from 3310 Oberwolfach Report 56/2018 which we want to extract average or macroscopic information, see [2, 5, 7, 10, 12] and the references therein. Some classical and modern instances of applications are: molecules in gases, electron transport in semiconductor materials, ions and electrons in plasmas, grains or beads in granular gases, stars or galaxies in astrophysics, endothelial cells in chemotactic movement for angiogenesis, neurons spike dynamics in neuroscience, fuel droplets in Diesel engines, dust particles in atmospheric pollution, animals in a swarm, agents in an economic market, pedestrians strolling around complex building geometries... This diversity of applications should not hide the common underlying methods, models and structural equations derived in all of these problems. Not only the beauty of applications but also the inherent mathematical difficulty of these models have attracted the attention of leading mathematicians from the modelling, analysis and numerical viewpoints as international scientific databases demonstrate. In all of these models, a fine study of the stability properties, the long-time asymptotics [12], the numerical methods and their simulation in the applications are relevant questions [7]. The individual behaviour of the “particles” is typically modelled via stochastic or deterministic ODEs from which one obtains mesoscopic descriptions based on kinetic type PDEs, while the average dynamics is usually described via continuum mechanics systems of hyperbolic, diffusive, or hydrodynamic type. The interplay between the longand short-range interactions, transport and diffusion, and their nonlocal and nonlinear features are the main mathematical difficulties in understanding equilibrium states, their stability and asymptotic analysis. On the other hand, the relation between kinetic equations and nonlinear nonlocal aggregation diffusion equations appears at the level of homogeneous kinetic models and Fokker-Planck type equations in which the exchange of different methods and techniques has recently provided important advances [1, 4, 8]. Hydrodynamic models are usually derived from kinetic equations via moment closures or via asymptotic limits. Nevertheless, plenty of challenging related questions remain at the hydrodynamic and kinetic description levels. Finally, the development of numerical schemes for both the microscopic and the kinetic level descriptions [7] faces the curse of the high dimensionality of the problems, not to mention the intricate structure of the convolution-like operators involved. The connection to macroscopic problems is then obtained through asymptotic limits, sometimes performed even at the level of the numerical schemes. These asymptotic preserving schemes are certainly a strategy to attack the reduction of computational cost at the kinetic level while keeping track of the microscopic dynamics if needed. On the other hand, numerical discretisations of macroscopic equations should reflect their structural properties. Therefore, one natural idea is to take advantage of the gradient flow structure in the macroscopic equations to construct numerical schemes based on calculus of variations or optimal transport viewpoint. The mini-workshop Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations, organized by Jose A. Carrillo (London), Martin Frank Innovative Trends for Kinetic Equations 3311 (Karlsruhe), Jingwei Hu (West Lafayette), and Lorenzo Pareschi (Ferrara) was well attended by 17 participants, including both junior and senior researchers and 3 females. We brought researchers working on various numerical aspects of kinetic modelling, their numerical analysis and their applications to exchange ideas and promote collaborations. Every participant contributed a talk which makes a total of 17 talks. In particular, there was a joint session on Thursday that gathered all the participants from the three mini-workshops held concurrently in the same week (all focused on numerics). Antoine Cerfon from our workshop gave a talk on kinetic simulation of plasmas in the joint session. Several topics on various numerical aspects of kinetic equations were addressed in the mini-workshop. Here is a brief summary. Uncertainty quantification (UQ) for kinetic equations. The kinetic equations often contain uncertainties in their collision kernels or scattering coefficients, initial or boundary data, forcing terms, geometry, etc. Quantifying the uncertainties in kinetic models have important engineering and industrial applications. There was a big emphasis in the mini-workshop on this topic. Specifically, the opening talk by Shi Jin conducted the sensitivity analysis for kinetic equations and used it to prove the spectral convergence of the stochastic Galerkin (sG) method. The talks by Giacomo Dimarco and Lorenzo Pareschi focused on the Monte Carlo (MC) method, where a control variate approach was introduced to reduce the variance of standard MC techniques. Finally, the talk by Mattia Zanella considered the UQ for kinetic equations arising in collective dynamics and a hybrid sG-MC method was presented. Structure-preserving methods for kinetic equations. Kinetic equations model the time evolution of the probability density function (PDF) and is usually endowed with an entropy functional. As such, the numerical methods that can preserve the properties of the solution, e.g., positivity, conservation, entropydecay, are highly-desirable. Furthermore, the kinetic equations are connected to macroscopic fluid equations as the Knudsen number (ratio of the mean free path and typical length scale) goes to zero. A numerical scheme that can capture the fluid limit without resolving the small scale, i.e., asymptotic-preserving (AP), is also attractive, especially for handling multiscale problems. Several talks in the mini-workshop focused on the design of structure-preserving methods for kinetic equations. Jingwei Hu introduced a time discretization method for a class of stiff kinetic equations that is both positivity-preserving and AP. Thomas Rey proposed a finite volume scheme for the linear kinetic equation that is able to capture the solution in long time and diffusive limit. Both Li Wang and Jose A. Carrillo studied a general nonlinear nonlocal Fokker-Planck type equation with a gradient flow structure: Li presented an optimization method based on Wasserstein metric and Jose presented a finite volume method. Both approaches are able to preserve the positivity and energy decay of the solution. Finally Giovanni Russo introduced a high-order semi-Lagrangian method for the BGK and Vlasov-Poisson equation whose key property is the conservation of mass. 3312 Oberwolfach Report 56/2018 Fast deterministic methods for kinetic equations. A prominent feature of many kinetic equations, for instance, the Boltzmann equation, is a nonlinear, nonlocal, high-dimensional collision operator. Numerically approximating these operators has been a big challenge in science and engineering for decades, and the Monte Carlo based stochastic method has been historically popular due its low complexity and simplicity. In recent years, the deterministic numerical methods (e.g., spectral method, discontinuous Galerkin (DG) method) have seen their revival as the computing power grows. We have two talks in our mini-workshop addressing the deterministic approximation of the Boltzmann type equations. The talk by Ralf Hiptmair reported the recent progress of constructing finite element approximation of the Boltzmann equation. The talk by Zheng Ma introduced a fast Fourier spectral method for the inelastic Boltzmann collision operator. Numerical methods in plasma physics. Kinetic description plays an important role in plasma physics where the underlying equation is the Vlasov-Poisson (or Maxwell) equations. If the collision is desired (for example, in hot plasmas), a term such as the Fokker-Planck or Landau operator would be added as well. This mini-workshop gathered both mathematicians and plasma physicists to discuss many challenging problems in plasma simulations. The talk by Francis Filbet addressed the asymptotics of the Vlasov equation in a strong magnetic field. The talk by Luis Chacon focused on the collisional plasmas. The talk by Antoine Cerfon introduced a sparse grid technique to speed up large scale computations. Other topics. A few other topics were als