Small Collaboration: Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations

Abstract

Nonlinear hyperbolic partial differential equations constitute a plethora of models from physics, biology, engineering, etc. In this workshop we cover the range from modeling, mathematical questions of well-posedness, numerical discretization and numerical simulations to compare with the phenomenon from nature that was modeled in the first place. Both kinetic and fluid models were discussed. Mathematics Subject Classification (2010): 35B40, 35L65, 35Q20, 35R30, 65M06, 76W05, 82C40. Introduction by the Organizers The workshop (held in a hybrid format) titled Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations, organized by Christian Klingenberg (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Marlies Pirner (Würzburg, Germany) was attended by 18 participants, 9 of which were female. Nonlinear hyperbolic systems of time-dependent partial differential equations are important mathematical models for a large number of complex natural systems of fundamental interest. The governing equations can be derived from first principles. In applications these models are used with great success. In this workshop we spanned the gamut from modeling physical phenomena by hyperbolic partial differential equations, discussing questions of existence, uniqueness and well-posedness, their numerical discretization and related numerical analysis and numerical implementation questions to numerical simulations in order to ascertain how this matches the physical phenomenon at hand. 2 Oberwolfach Report 19/2021 Depending on the time scale and spatial scale in the application at hand, the models may be either microscopic kinetic equations or a macroscopic fluid equations. Both types of models were discussed. Next we list some of those topics. Kinetic models Here we take an atomistic view of the flow. Considering density distributuons of these micrsoscopically interacting particles, we obtain Boltzmann-type equations consisting of a first order transport operator for the density distribution of the particles which are set equal to a zeroth order term describing the interaction between the particles. As reported on by Marlies Pirner, she models gas mixtures in this way, see e.g. [1], and proves that this model satisfies physical properties. The numerical implementation of this new model needs to be found. Seok-Bae Yun and Gi-Chan Bae reported on theoretical aspects of certain kinetic models, which lend itself to efficient numerical simulations, while still modeling the physics appropriately. For a kinetic model of plasma, the Vlasov-Poisson model, enhanced by a BGK relaxation term, aspects of Landau damping were discussed by Lena Baumann. The study of uncertainty quantification for kinetic models was discussed, see here [2]. But uncertainty quantification does not always represent the viewpoint of the experimentalists. Instead they want to determine the uncertain coefficient in a PDE by measuring the solution at the parts of the boundary, given data on other parts of the boundary. The lectures of Ru-Yu Lai introduced and reported new results on this subject. In other words experimentalists are interested in solving the inverse problem in a Bayesian setting, see here [3] for a related question. We followed on from this by considering a model from mathematical biology, namely the motion of cells, as described by the kinetic chemotaxis equations. The corresponding macroscopic Keller-Segel type model will be a diffusion equation. This was reported on by Min Tang. The aim is to study the inverse problems for these two settings. Kathrin Hellmuth reported on these ideas, see [11]. Numerical methods for kinetic equations that continue to be valid in the limit of Knudsen number going to zero are called asymptotic preserving. In addition we found numerical methods that at the same preserve stationary solutions, see here [4], as repoted on by Farah Kanbar. It was discussed on how to extend this to kinetic models of gas mixtures. Finally kinetic models for multi-species quantum particles are devised and existence is proven see here [5]. It was discussed how this can be translated to numerical schemes. This was reported on by Sandra Warnecke. The Euler equations of compressible gas dynamics, theory In one space dimension one has an understanding of existence and uniqueness of solutions to the compressible Euler equations. In that situation (under appropriate assumptions) a sequence of approximations to the Euler equations converges to the weak solutions of Euler equations. In higher space dimensions much less is understood, see here [6] and here [7]. Simon Markfelder reported on this circle of Modeling Phenomena from Nature by Hyperbolic PDEs 3 ideas. Most likely weak solutions are not the appropriate solution concept here. The goal is to identify a proper notion for solutions. Eduard Feireisl studied this in two lectures in the context of stochastics. Solution concepts of the compressible multi-dimensional Euler equations may also be studied by looking at limits of numerical approximations. This was reported on by Eva Horlebein. The Euler equations of compressible gas dynamics, numerics Numerics of conservation laws has been dominated by the idea of Godunov, where a crucial ingredient has been the propagation of discontinuous data, the Riemann problem. This is a one-space-dimensional idea. It seems it is time for a change of paradigm. Following a suggestion of Phil Roe, in multiple space dimensions the evolution of continuous finite elements by using (almost) exact evolution at discrete points seems to be very promising, see here [8]. The study of such genuinely multidimensional schemes holds enormous promise, because they naturally satisfy all involution constraints. Progress towards this was presented by Wasilij Barsukov. For the Euler equations with gravity we seek numerical methods that are both asymptotic preserving in the low Mach limit and also stationary preserving, also know as well-balanced, see here [9], or here [10]. Ways on how to improve numerical schemes that combine these two features were reported on by Claudius Birke and Philipp Edelmann. Overall this workshop gave space to discuss the above circle of ideas in the wonderful atmosphere of Oberwolfach. References [1] J. Haack, C. Hauck, C. Klingenberg, M. Pirner, S. Warnecke, A consistent BGK model with velocity-dependent collision frequency for gas mixtures, submitted 2021, see paper here [2] Herzing, T., Klingenberg, C., Pirner, M.: Hypocoercivity of the linearized BGK-equation with stochastic coefficients, submitted (2020), see paper here [3] Klingenberg, C., Lai, R., Li, Q.: Reconstruction of the emission coefficient in the nonlinear radiative transfer equation, SIAM Journal on Applied Mathematics, Vol. 81, 1 (2021) see the paper here [4] Emako, F. Kanbar, C. Klingenberg, M. Tang, A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving, submitted (2021) see paper here [5] G. Bae, C. Klingenberg, M. Pirner. S. Yun: BGK model of the multi-species Uehling Uhlenbeck equation, Kinetic and Related Models, Vol. 14, Issue 1 (2021) see paper here [6] Feireisl, E., Klingenberg, C., Kreml, O., Markfelder, S.: On oscillatory solutions to the complete Euler equations, Journal of Differential Equations, Vol. 296, issue 2, pp 1521-1543, (2020), see paper here [7] E. Feireisl; C. Klingenberg; S. Markfelder, ’On the density of wild initial data for the compressible Euler system’, Calculus of Variations (2020), see paper here [8] Wasilij Barsukow; Jonathan Hohm; Christian Klingenberg; Philip L Roe, The active flux scheme on Cartesian grids and its low Mach number limit, Journal of Scientific Computing, vol. 81, pp. 594-622, (2019), see paper here [9] Berberich, J., Käppeli, R., Chandrashekar, P., Klingenberg, C.: High order discretely wellbalanced methods for arbitrary hydrostatic atmospheres, Communications in Computational Physics (2021), see paper here 4 Oberwolfach Report 19/2021 [10] Edelmann, Horst, Berberich, Andrassy, Higl, Klingenberg, Röpke: Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers, submitted 2021 see paper here [11] Helmuth, K., Klingenberg, C., Li, Q., Tang, M.: Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting, in the volume Inverse Problems with Partial Data edited by Qin Li and Li Wang, submitted to Computation (2021) Acknowledgement: We thank the MFO for generously supporting this workshop by awarding OWLG grants in addition to VCA grants to on-site participants of this workshop. Modeling Phenomena from Nature by Hyperbolic PDEs 5 Small Collaboration (hybrid meeting): Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations