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In this paper, we present high-performance computing for the BGK model of the Boltzmann equations with a meshfree method. We use the Arbitrary-Lagrangian-Eulerian (ALE) method, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method. We have used the Graphics Processing Unit (GPU) to accelerate the code and compared with the CPU code. Two and three dimensional driven cavity problems are solved, where we have obtained the speed up up to 307 times and 127 times in two and three dimensional cases, respectively., Comment: arXiv admin note: text overlap with arXiv:2109.05822

In the world of epidemics, the mathematical modeling of disease co-infection is gaining importance due to its contributions to mathematics and public health. Because the co-infection may have a double burden on families, countries, and the universe, understanding its dynamics is paramount. We study a SEIQR (susceptible-exposed-infectious-quarantined-recovered) deterministic epidemic model with a single host population and multiple strains (-$c$ and -$i$) to account for two competitive diseases with quarantine effects. To model the role of quarantine and isolation efficacy in disease dynamics, we utilize a linear function. Further, we shed light on the standard endemic threshold and determine the conditions for extinction or coexistence with and without forming co-infection. Next, we show the dependence of the criticality based on specific parameters of the different pathogens. We found that the disease-free equilibrium (DFE) of the single-strain model always exists and is globally asymptotically stable (GAS) if $\tilde{\mathcal{R}}_k^q\leq 1$, else, a stable endemic equilibrium. On top of that, the model has forward bifurcation at $\tilde{\mathcal{R}}_k^q = 1$. In the case of a two-strain model, the strain with a large reproduction number outcompetes the one with a smaller reproduction number. Further, if the co-infected quarantine reproduction number is less than one, the infections of already infected individuals will die out, and co-infection will persist in the population otherwise. We note that the quarantine and isolation of exposed and infected individuals will reduce the number of secondary cases below one, consequently reducing the disease complications if the total number of people in the quarantine is at most the critical value.

Starting from a non-local version of the Prigogine-Herman traffic model, we derive a natural hierarchy of kinetic discrete velocity models for traffic flow consisting of systems of quasi-linear hyperbolic equations with relaxation terms. The hyperbolic main part of these models turns out to have several favourable features. In particular, we determine Riemann invariants and prove richness and total linear degeneracy of the hyperbolic systems. Moreover, a physically reasonable invariant domain is obtained for all equations of the hierarchy. Additionally, we investigate the full relaxation system with respect to stability and persistence of periodic (stop and go type) solutions and derive a condition for the appearance of such solutions. Finally, numerical results for various situations are presented, illustrating the analytical findings.

Based on the continuum model for granular media developed in Dunatunga et al. we propose a mesh-free generalized finite difference method for the simulation of granular flows. The model is given by an elasto-viscoplastic model with a yield criterion using the $\mu(I)$ rheology from Jop et al. The numerical procedure is based on a mesh-free particle method with a least squares approximation of the derivatives in the balance equations combined with the numerical algorithm developed in Dunatunga et al. to compute the plastic stresses. The method is numerically tested and verified for several numerical experiments including granular column collapse and rigid body motion in granular materials. For comparison a nonlinear microscopic model from Lacaze et al. is implemented and results are compared to the those obtained from the continuum model for granular column collapse and rigid body coupling to granular flow.

In this paper we present a novel technique for the simulation of moving boundaries and moving rigid bodies immersed in a rarefied gas using an Eulerian-Lagrangian formulation based on least square method. The rarefied gas is simulated by solving the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation of rarefied gas dynamics. The BGK model is solved by an Arbitrary Lagrangian-Eulerian (ALE) method, where grid-points/particles are moved with the mean velocity of the gas. The computational domain for the rarefied gas changes with time due to the motion of the boundaries. To allow a simpler handling of the interface motion we have used a meshfree method based on a least-square approximation for the reconstruction procedures required for the scheme. We have considered a one way, as well as a two-way coupling of boundaries/rigid bodies and gas flow. The numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. Convergence studies are performed for one-dimensional and two-dimensional test-cases. Several further test problems and applications illustrate the versatility of the approach.

In this paper we propose coupling conditions for a relaxation model for vehicular traffic on networks. We present a matched asymptotic expansion procedure to derive a LWR- network with well-known classical coupling conditions from the relaxation network in the macroscopic limit. Similar to the asymptotic limit of boundary value problems, we perform an asymptotic analysis of the interface layers at the nodes and a matching procedure using half-Riemann problems for the limit conservation law. Moreover, we present numerical experiments comparing the relaxation network with the LWR network for a broader range of coupling conditions., Comment: arXiv admin note: substantial text overlap with arXiv:2002.05995

Modeling and simulation of disease spreading in pedestrian crowds has been recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts also the influence of contact times has been modelled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated including uni- and bi-directional flow in a passage with and without obstacles.

In this paper we propose coupling conditions for a kinetic two velocity model for vehicular traffic for junctions with diverging lanes. We consider cases with and without directional preferences and present corresponding kinetic coupling conditions. From this kinetic network model coupling conditions for a macroscopic traffic model are derived. We use an analysis of the layer equations at the junction in combination with a suitable matching procedure with half-Riemann problems for the macroscopic model. In this way classical coupling conditions for scalar conservation laws for traffic flow on networks are derived from an underlying network problem., Comment: arXiv admin note: substantial text overlap with arXiv:2002.05995

In this paper we propose coupling conditions for a kinetic two velocity model for vehicular traffic on networks. These conditions are based on the consideration of the free space on the respective roads. The macroscopic limit of the kinetic relaxation system is a classical scalar conservation law for traffic flow. Similar to the asymptotic limit of boundary value problems for kinetic models, we consider here the limit of the full network problem including the coupling conditions at the nodes. An asymptotic analysis of the interface layers at the nodes and a matching procedure using half-Riemann problems for the limit conservation law are used to derive coupling conditions for classical macroscopic traffic models on the network from the kinetic ones.

We study the optimal control of district heating networks using a reduced order model based on a system theoretic description close to the underlying Euler equations. In the presented scenarios, the central task is to limit the maximal feed-in power occurring as a product of control and state variables. The underlying dynamics of heating networks acting as optimization constraints pose the central computational complexity, prohibiting the determination of an optimal control online. The advection of the injected energy density on the network results in an index-1, quadratic in state differential algebraic equation, challenging to reduce. The suggested reduced model decreases the computation time of the optimization significantly. The effectiveness of the presented approach is demonstrated for an existing, large-scale heating network including changes of flux directions.

In this article a framework for the generation of a computationally fast surrogate model for district heating networks is presented. An appropriate model results in an index-1 hyperbolic, differential algebraic equation quadratic in state, exhibiting several hundred of outputs to be approximated. We show the existence of a global energy matrix which fulfills the Lyapunov inequality ensuring stability of the reduced model. By considering algebraic variables as parameters to the dynamical transport, the reduction of a linear, time varying (LTV) problem results. We present a scheme to efficiently combine linear reductions to a global surrogate model using a greedy strategy in the frequency domain. The numerical effectiveness of the scheme is demonstrated at different, existing, large scale networks.

In this paper we have applied a Semi-Lagrangian schemes with meshfree interpolation, based on a Moving Least Squares (MLS) method, to solve the BGK model for rarefied gas dynamics. Sod's shock tube problems are presented for a large range of mean free paths in one dimensional physical space and three dimensional velocity space. In order to validate the solutions obtained from the meshfree method, we have used the piecewise linear spline interpolation. Furthermore, we have compared the solutions of the BGK model with the solutions obtained from Direct Simulation Monte Carlo (DSMC) method. In the case of a very small mean free path the numerical solutions are compared with the exact solutions of the compressible Euler equations. Overall we found that the meshfree interpolation gives better approximation than the piecewise linear spline interpolation.

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)

The force and torque on a Janus sphere moving in a rarefied gas with a thermal gradient are calculated. The regime of large Knudsen number is considered, with the momenta of impinging gas molecules either obtained from a Chapman-Enskog distribution or from a binary Maxwellian distribution between two opposing parallel plates at different temperature. The reflection properties at the surface of the Janus particle are characterized by accommodation coefficients having constant but dissimilar values on each hemisphere. It is shown that the Janus particle preferentially orients such that the hemisphere with a larger accommodation coefficient points towards the lower temperature. The thermophoretic velocity of the particle is computed, and the influence of the thermophoretic motion on the magnitude of the torque responsible for the particle orientation is studied. The analytical calculations are supported by Direct Simulation Monte Carlo results, extending the scope of the study towards smaller Knudsen numbers. The results shed light on the efficiency of oriented deposition of nanoparticles from the gas phase onto a cold surface.

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.

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.

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.

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., Comment: 20 pages, 12 figures. arXiv admin note: text overlap with arXiv:1406.4734

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.

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.

Simulating the detailed movement of a rising bubble can be challenging, especially when it comes to bubble path instabilities. Free rising ellipsoidal bubbles not only move in straight lines but can describe sinusoidal, zigzag or spiraling paths. The common Euler Euler (EE) simulation techniques can no longer resolve the actual movement patterns and Direct Numerical Simulations (DNS) tend to be very costly when simulating a larger number of bubbles. A solution based on the Euler Lagrange (EL) approach is presented, where the bubbles show oscillating shape and/or instable paths while computational cost are at a far lower level than in DNS. The model calculates direction, shape and rotation of the bubbles to finally create typical instable path lines. This is embedded in an EL simulation, which can resolve bubble size distribution, mass transfer and chemical reactions. To ensure realistic solution, validation against experimental data of single rising bubbles and bubble swarms are presented. References with 2D and also 3D analysis are taken into account to compare simulative data in terms of typical geometrical parameters and average field values., Comment: 18 pages, 11 figures, to be submitted to the Chemical Engineering Journal

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.

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., Comment: International Journal for Numerical Methods in Engineering

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.

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.

In this paper we consider kinetic and associated macroscopic models for chemotaxis on a network. Coupling conditions at the nodes of the network for the kinetic problem are presented and used to derive coupling conditions for the macroscopic approximations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations asymptotic preserving relaxation schemes are extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

In this paper, a delay Vlasov-Fokker-Planck equation associated to a stochastic interacting particle system with delay is investigated analytically. Under certain restrictions on the parameters well-posedness and ergodicity of the mean-field equation are shown and an exponential rate of convergence towards the unique stationary solution is proven as long as the delay is finite. For infinte delay i.e., when all the history of the solution paths are taken into consideration polynomial decay of the solution is shown.

We consider a mean field hierarchy of models for large systems of interacting ellipsoids suspended in an incompressible fluid. The models range from microscopic to macroscopic mean field models. The microscopic model is based on three ingredients. Starting from a Langevin type model for rigid body interactions, we use a Jefferys type term to model the influence of the fluid on the ellipsoids and a simplified interaction potential between the ellipsoids to model the interaction between the ellipsoids. A mean field equation and corresponding equations for the marginals of the distribution function are derived and a numerical comparison between the different levels of the model hierarchy is given. The results clearly justify the suitability of the proposed approximations for the example cases under consideration.

Reduced one-dimensional equations for the stationary, isothermal rotational spinning process of slender fibers are considered for the case of large Reynolds ($\delta=3/\text{Re}\ll 1$) and small Rossby numbers ($\varepsilon \ll 1$). Surface tension is included in the model using the parameter $\kappa=\sqrt{\pi}/(2 \text{We})$ related to the inverse Weber number. The inviscid case $\delta=0$ is discussed as a reference case. For the viscous case $\delta > 0$ numerical simulations indicate, that for a certain parameter range, no physically relevant solution may exist. Transferring properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for $\delta > 3\varepsilon^2 \left( 1- \frac{3}{2}\kappa +\frac{1}{2}\kappa^2\right)$ no physical relevant stationary solution can exist., Comment: 9 pages, 4 figures

This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e. low-Mach-number--slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a spatial Gauss-Legendre collocation and an $\alpha$-method in time.

We consider an interacting system of one-dimensional structures modelling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic and macroscopic) is included., Comment: 23 pages, 16 figures

In this paper we present a numerical scheme to simulate a moving rigid body with arbitrary shape suspended in a rarefied gas. The rarefied gas is simulated by solving the Boltzmann equation using a DSMC particle method. The motion of the rigid body is governed by the Newton-Euler equations, where the force and the torque on the rigid body is computed from the momentum transfer of the gas molecules colliding with the body. On the other hand, the motion of the rigid body influences the gas flow in its surroundings. We validate the numerical results by testing the Einstein relation for Brownian motion of the suspended particle. The translational as well as the rotational degrees of freedom are taken into account. It is shown that the numerically computed translational and rotational diffusion coefficients converge to the theoretical values., Comment: 16 pages, 8 figures

Simulations of wetting phenomena by a meshfree particle method are presented. The incompressible Navier-Stokes equations are used to model the two-phase flow. The continuous surface force model is used to incorporate the surface tension force. Chorin's projection method is applied to discretize the Navier-Stokes equations. The different fluid phases are identified by assigning different colors and different material properties (density, viscosity) to the particles that remain unchanged throughout a simulation. Two-phase flow is captured by a one-fluid model via using weighted averages of the density and viscosity in a region around the fluid-fluid interface. The differential operators at each particle are computed from the surrounding cloud of particles with the help of the least-squares method. The numerical results are compared with specific analytical solutions, but also with previously considered test cases involving wetting of a container and sessile drops. A good overall agreement is found., Comment: 34 pages, 41 figures

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum entropy Mn models. Realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closures. They provide non-negative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite-difference Fokker-Planck scheme.

A meshfree Lagrangian method for the fluctuating hydrodynamic equations (FHEs) with fluid-structure interactions is presented. Brownian motion of the particle is investigated by direct numerical simulation of the fluctuating hydrodynamic equations. In this framework a bidirectional coupling has been introduced between the fluctuating fluid and the solid object. The force governing the motion of the solid object is solely due to the surrounding fluid particles. Since a meshfree formulation is used, the method can be extended to many real applications involving complex fluid flows. A three-dimensional implementation is presented. In particular, we observe the short and long-time behaviour of the velocity autocorrelation function (VACF) of Brownian particles and compare it with the analytical expression. Moreover, the Stokes-Einstein relation is reproduced to ensure the correct long-time behaviour of Brownian dynamics., Comment: 24 pages, 2 figures

In this paper we propose a model for a sewer network coupled to surface flow and investigate it numerically. In particular, we present a new model for the manholes in storm sewer systems. It is derived using the balance of the total energy in the complete network. The resulting system of equations contains, aside from hyperbolic conservation laws for the sewer network and algebraic relations for the coupling conditions, a system of ODEs governing the flow in the manholes. The manholes provide natural points for the interaction of the sewer system and the run off on the urban surface modelled by shallow water equations. Finally, a numerical method for the coupled system is presented. In several numerical tests we study the influence of the manhole model on the sewer system and the coupling with 2D surface flow.