arXiv subjects

Raul Borsche

Publications and source records attributed to Raul Borsche.

At least 19 recordsLinked to original sources

arXiv2026

A spectral approach to interface layers on networks for the linearized BGK equation and its acoustic limit

We consider in this paper a velocity discretized version of the full linear kinetic BGK model and the corresponding limit for small Knudsen number, the linearised Euler or acoustic system. Considering these equations on networks, coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. Here, a degeneracy in the limit equations requires not only the investigation of kinetic layers, but also the discussion of viscous layers. Using the kinetic coupling conditions at the junction and coupling kinetic and viscous layers to the outer problems on the edges one obtains a coupled kinetic half-space problem at each node. A spectral method is developed to solve this coupled kinetic half-space problems. This allows to obtain a detailed picture of the various interface layers near the nodes and to determine the relevant coefficients in the kinetic derived coupling conditions for the macroscopic equations.Numerical results show the accuracy and efficiency of the approach.

math.AP↗

arXiv2025

Interface layers and coupling conditions for discrete kinetic models on networks: a spectral approac

We consider kinetic and related macroscopic equations on networks. A class of linear kinetic BGK models is considered, where the limit equation for small Knudsen numbers is given by the wave equation. Coupling conditions for the macroscopic equations are obtained from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network and the consideration of coupled solutions of kinetic half-space problems. Analytical results are obtained for a discrete velocity version of the coupled half-space problems. Moreover, an efficient spectral method is developed to solve the coupled discrete velocity half-space problems. In particular, this allows to determine the relevant coefficients in the coupling conditions for the macroscopic equations from the underlying kinetic network problem. These coefficients correspond to the so-called extrapolation length for kinetic boundary value problems. Numerical results show the accuracy and fast convergence of the approach. Moreover, a comparison of the kinetic solution on the network with the macroscopic solution is presented.

math.NA↗

arXiv2025

2D hydrodynamic simulation of TeraFETs beyond the gradual-channel approximation for transient, large-signal or ultrahigh-frequency simulations

In the past decade, detection of THz radiation by plasma-wave-assisted frequency mixing in antenna-coupled field-effect transistors (TeraFETs) -- implemented in various semiconductor material systems (Si CMOS, GaN/AlGaN, GaAs/AlGaAs, graphene, etc.) -- has matured and led to a practically applied detector technology. This has been supported by the development of powerful device simulation tools which take into account relevant collective carrier dynamics and mixing processes in various approximations. These tools mostly model carrier transport in 1D and they are usually geared towards continuous-wave illumination of the device and small-signal response. Depending on their implementation, it may not be possible readily to simulate large-signal and pulsed operation. Another approximation which may lead to unsatisfactory results is the 1D restriction to calculate only the longitudinal electric field components. Especially at the edges of the gate electrode, solving of the 2D Poisson equation promises better results. This contribution introduces a stable way to solve the 2D Poisson equation self-consistently with the hydrodynamic transport equations including the numerically challenging convection term. We employ a well-balanced approximate Harten-Lax-van-Leer-Contact Riemann solver. The approach is well suited for a future treatment of transient and large-signal cases. The 2D treatment also generically extends the model beyond the gradual-channel approximation and allows to calculate the FET's response at high THz frequencies where the gate-to-channel potential acquires a non-local character. Model calculations are performed for the exemplary case of a 65-nm Si CMOS TeraFET in the isothermal approximation.

physics.comp-ph↗

arXiv2023

Implicit Active Flux methods for linear advection

In this work we develop implicit Active Flux schemes for the scalar advection equation. At every cell interface we approximate the solution by a polynomial in time. This allows to evolve the point values using characteristics and to update the cell averages using fluxes obtained by integrating this polynomial. The resulting schemes have order of convergence up to five, but show only moderate oscillations with high frequencies for discontinuous solutions. In numerical experiments we compare the different methods and show an application to network flows.

math.NA↗

arXiv2023

A hierarchy of kinetic discrete-velocity models for traffic flow derived from a non-local Prigogine-Herman model

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.

math.NA↗

arXiv2021

Relaxation models for scalar traffic networks and zero relaxation limit

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.

math.AP↗

arXiv2020

A kinetic traffic network model and its macroscopic limit: diverging lanes

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.

math.AP↗

arXiv2020

A kinetic traffic network model and its macroscopic limit: merging lanes

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.

math.AP↗

arXiv2018

Kinetic layers and coupling conditions for nonlinear scalar equations on networks

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.

math.AP↗

arXiv2017

A nonlinear discrete-velocity relaxation model for traffic flow

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.

math.AP↗

arXiv2017

Kinetic layers and coupling conditions for macroscopic equations on networks I: the wave equation

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.

math.AP↗

arXiv2016

Numerical Approximation of Hyperbolic Systems Containing an Interface

In this paper we present an approach to approximate numerically the solution of coupled hyperbolic conservation laws. The coupling is achieved through a fixed interface, in which interface conditions are linking the traces of both sides. The numerical solver is based on central methods, like the Rusanov scheme, and does not use the structure of the Riemann Problem. It consists in balancing the effects of the waves that enter the interface. The scheme is well balanced with respect to all the piecewise constant equilibria associated with the interface condition and is able to maintain exactly conservation properties of the interface conditions. A detailed analysis and several numerical tests show the quality of the method. Different applications, including sonic and transsonic flows and a multiphysic model are studied.

math.NA↗

arXiv2015

Kinetic and related macroscopic models for chemotaxis on networks

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.

math.AP↗

arXiv2015

Mean field models for interacting ellipsoidal particles

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.

math.DS↗

arXiv2015

High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models

In this paper we construct high order finite volume schemes on networks of hyperbolic conservation laws with coupling conditions involving ODEs. We consider two generalized Riemann solvers at the junction, one of Toro-Castro type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is treated with a Taylor method or an explicit Runge-Kutta scheme, respectively. Both resulting high order methods conserve quantities exactly if the conservation is part of the coupling conditions. Furthermore we present a technique to incorporate lumped parameter models, which arise from simplifying parts of a network. The high order convergence and the robust capturing of shocks is investigated numerically in several test cases.

math.NA↗

arXiv2015

A Retarded Mean-Field Approach for Interacting Fiber Structures

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.

math.DS↗

arXiv2014

Differential Equations Modeling Crowd Interactions

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an "ad hoc" well posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting non locally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians, and also with other "agents".

math.AP↗

arXiv2014

Flooding in urban drainage systems: Coupling hyperbolic conservation laws for sewer systems and surface flow

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.

physics.flu-dyn↗