Axel Klar

Pratik Suchde, Joerg Kuhnert, Simon Schroeder et al.

Lack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate conservation of numerical fluxes. This numerical flux conservation is done within the usual moving least squares framework. Unlike Finite Volume Methods, it is based on locally defined control cells, rather than a globally defined mesh. We present the application of this method to an advection diffusion equation and the incompressible Navier - Stokes equations. Our simulations show that the introduction of flux conservation significantly reduces the errors in conservation in meshfree GFDMs.

Raul Borsche, Mark Kimathi, Axel Klar

In the present paper a review and numerical comparison of a special class of multi-phase traffic theories based on microscopic, kinetic and macroscopic traffic models is given. Macroscopic traffic equations with multi-valued fundamental diagrams are derived from different microscopic and kinetic models. Numerical experiments show similarities and differences of the models, in particular, for the appearance and structure of stop and go waves for highway traffic in dense situations. For all models, but one, phase transitions can appear near bottlenecks depending on the local density and velocity of the flow.

Naveen K. Mahato, Axel Klar, Sudarshan Tiwari

We consider a multi-group microscopic model for pedestrian flow describing the behaviour of large groups. It is based on an interacting particle system coupled to an eikonal equation. Hydrodynamic multi-group models are derived from the underlying particle system as well as scalar multi-group models. The eikonal equation is used to compute optimal paths for the pedestrians. Particle methods are used to solve the equations on all levels of the hierarchy. Numerical test cases are investigated and the models and, in particular, the resulting evacuation times are compared for a wide range of different parameters.

Gregor Corbin, Alexander Hunt, Florian Schneider et al.

Starting from a two-scale description involving receptor binding dynamics and a kinetic transport equation for the evolution of the cell density function under velocity reorientations, we deduce macroscopic models for glioma invasion featuring partial differential equations for the mass density and momentum of a population of glioma cells migrating through the anisotropic brain tissue. The proposed first and higher order moment closure methods enable numerical simulations of the kinetic equation. Their performance is then compared to that of the diffusion limit. The approach allows for DTI-based, patient-specific predictions of the tumor extent and its dynamic behavior.

Raul Borsche, Axel Klar

We derive a nonlinear 2-equation discrete-velocity model for traffic flow from a continuous kinetic model. The model converges to scalar Lighthill-Whitham type equations in the relaxation limit for all ranges of traffic data. Moreover, the model has an invariant domain appropriate for traffic flow modeling. It shows some similarities with the Aw-Rascle traffic model. However, the new model is simpler and yields, in case of a concave fundamental diagram, an example for a totally linear degenerate hyperbolic relaxation model. We discuss the details of the hyperbolic main part and consider boundary conditions for the limit equations derived from the relaxation model. Moreover, we investigate the cluster dynamics of the model for vanishing braking distance and consider a relaxation scheme build on the kinetic discrete velocity model. Finally, numerical results for various situations are presented, illustrating the analytical results.

Juliane Ritter, Axel Klar, Florian Schneider

The aim of this work is to investigate the application of partial moment approximations to kinetic chemotaxis equations in one and two spatial dimensions. Starting with a kinetic equation for the cell densities we apply a half-/quarter-moments method with different closure relations to derive macroscopic equations. Appropriate numerical schemes are presented as well as numerical results for several test cases. The resulting solutions are compared to kinetic reference solutions and solutions computed using a full moment method with a linear superposition strategy.

Raul Borsche, Axel Klar

We consider a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic equation on a network. Coupling conditions for the macroscopic equations are derived from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the combination of kinetic half-space problems with Riemann problems at the junction. Detailed numerical comparisons between the different models show the agreement of the coupling conditions for the case of tripod networks.

Sudarshan Tiwari, Axel Klar, Steffen Hardt

We study the dynamics of a liquid droplet inside a gas over a large range of the Knudsen numbers. The moving liquid droplet is modeled by the incompressible Navier-Stokes equations, the surrounding rarefied gas by the Boltzmann equation. The interface boundary conditions between the gas and liquid phases are derived. The incompressible Navier-Stokes equations are solved by a meshfree Lagrangian particle method called Finite Pointset Method (FPM), and the Boltzmann equation by a DSMC type of particle method. To validiate the coupled solutions of the Boltzmann and the incompressible Navier-Stokes equations we have further solved the compressible and the incompressible Navier-Stokes equations in the gas and liquid phases, respectively. In the latter case both the compressible and the incompressible Navier-Stokes equations are also solved by the FPM. In the continuum regime the coupled solutions obtained from the Boltzmann and the incompressible Navier-Stokes equations match with the solutions obtained from the compressible and the incompressible Navier-Stokes equations. In this paper, we presented solutions in one-dimensional physical space.

Axel Klar, Sudarshan Tiwari

A multi-scale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for large and small interaction radii. The well resolved case for large interaction radius is treated, as well as underresolved situations with small values of the interaction radius. In the present work we extend the approach from [39] for porous media type limit equations to a more general case, including in particular hyperbolic limits. The method can be viewed as a numerical transition between a DEM-type method for microscopic interacting particle systems and a meshfree particle method for macroscopic equations. We discuss in detail the numerical performance of the scheme for various examples and the potential gain in computation time. The latter is shown to be particularly high for situations near the macroscopic limit. There are various applications of the method to problems involving mean field approximations in swarming, tra?c, pedestrian or granular fow simulation.

Raul Borsche, Axel Klar, Mark Kimathi

Kinetic models for vehicular traffic are reviewed and considered from the point of view of deriving macroscopic equations. A derivation of the associated macroscopic traffic flow equations leads to different types of equations: in certain situations modified Aw-Rascle equations are obtained. On the other hand, for several choices of kinetic parameters new Hamilton-Jacobi type traffic equations are found. Associated microscopic models are discussed and numerical experiments are presented discussing several situations for highway traffic and comparing the different models.

Klaas Willems, Axel Klar, Giovanni Russo et al.

We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The gas is described by the BGK equation in Lagrangian form and solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where the domain boundaries are time-dependent and the simulation domain contains rigid objects. Due to the irregular nature of the grid, we use a novel meshless MUSCL-like Moving Least Squares Method (MLS) for spatial discretisation coupled with a higher-order Implicit-Explicit Runge-Kutta method. To avoid spurious oscillations at discontinuities, we use the so-called Multi-dimensional Optimal Order Detection (MOOD) method with an adapted criterion to relax the discrete maximum property. Finally, we employ a new implementation of the boundary conditions that requires no iterative or extrapolation procedure. The method achieves fourth-order in 1D and second-order in 2D for simulations with moving boundaries. We demonstrate the method's effectiveness on classical test cases such as the driven square cavity, shear layer, and shock tube.

Naveen Kumar Mahato, Axel Klar, Sudarshan Tiwari

We analyze numerically some macroscopic models of pedestrian motion such as Hughes model [1] and mean field game with nonlinear mobilities [2] modeling fast exit scenarios in pedestrian crowds. A model introduced by Hughes consisting of a non-linear conservation law for the density of pedestrians coupled with an Eikonal equation for a potential modeling the common sense of the task. Mean field game with nonlinear mobilities is obtained by an optimal control approach, where the motion of every pedestrian is determined by minimizing a cost functional, which depends on the position, velocity, exit time and the overall density of people. We consider a parabolic optimal control problem of nonlinear mobility in pedestrian dynamics, which leads to a mean field game structure. We show how optimal control problem related to the Hughes model for pedestrian motion. Furthermore we provide several numerical results which relate both models in one and two dimensions. References [1] Hughes R.L.: A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36, 507-535 (2000) [2] Burger M., Di Francesco M., Markowich P.A., Wolfram M-T.: Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 19, 1311-1333 (2014)

Laura Müller, Axel Klar, Florian Schneider

We investigate the application of the method of moments approach for the one-dimensional population balance equation. We consider different types of moment closures, namely polynomial (P_N) closures, maximum entropy (M_N) closures and the quadrature method of moments QMOM_N. Realizability issues and implementation details are discussed. The numerical examples range from spatially homogeneous cases to a population balance equation coupled with fluid dynamic equations for a lid-driven cavity test case. A detailed numerical discussion of accuracy, order of the moment method and computational time is given.

Mark W. Hlawitschka, Sudarshan Tiwari, James Kwizera et al.

In this paper we present the comparison of experiments and numerical simulations for bubble cutting by a wire. The air bubble is surrounded by water. In the experimental setup an air bubble is injected on the bottom of a water column. When the bubble rises and contacts the wire, it is separated into two daughter bubbles. The flow is modeled by the incompressible Navier-Stokes equations. A meshfree method is used to simulate the bubble cutting. We have observed that the experimental and numerical results are in very good agreement. Moreover, we have further presented simulation results for liquid with higher viscosity. In this case the numerical results are close to previously published results.

Raul Borsche, Axel Klar

We consider kinetic and associated macroscopic equations on networks. The general approach will be explained in this paper for a linear kinetic BGK model and the corresponding limit for small Knudsen number, which is the wave equation. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.

Anamika Pandey, Axel Klar, Sudarshan Tiwari

In the current study a meshfree Lagrangian particle method for the Landau-Lifshitz Navier-Stokes (LLNS) equations is developed. The LLNS equations incorporate thermal fluctuation into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation theorem. The study focuses on capturing the correct variance and correlations computed at equilibrium flows, which are compared with available theoretical values. Moreover, a numerical test for the random walk of standing shock wave has been considered for capturing the shock location.