Abstract
While macroscopic traffic flow models adopt a fluid dynamic description of traffic, microscopic traffic flow models describe the dynamics of individual vehicles. Capturing macroscopic traffic phenomena accurately remains a challenge for microscopic models, especially in complex road sections. Based on a macroscopic network flow model calibrated to real traffic data and new rules for the acceleration and merging behavior on the on-ramp, we propose a microscopic model for on-ramps. To evaluate the performance of the new flow-based model, we conduct traffic simulations assessing speeds, accelerations, lane change positions, and risky behavior. Our results show that, although the proposed model exhibits some limitations, its performance is superior to the Intelligent Driver Model in the evaluated aspects. While the Intelligent Driver Model simulations are almost free of conflicts, the proposed model evokes a realistic amount and severity of conflicts and therefore can be considered for safety analysis.
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Abstract
A new coupling rule for the Lighthill–Whitham—Richards model at mergingjunctions is introduced that imposes the preservation of the ratio between inflowfrom a given road to the total inflow into the junction. This rule is consideredboth in the context of the original traffic flow model and a relaxation setting giving rise to two different Riemann solvers that are discussed for merging 2-to-1 junctions. Numerical experiments are shown, suggesting that the relaxation based Riemann solver is capable of suitable predictions of both free-flow andcongestion scenarios without relying on flow maximization.
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arXiv:2405.21005
Proceedings in Applied Mathematics and MechanicsVolume 24, Issue 4: Special Issue: 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) Dec 2024
Abstract
This work introduces a reduced one-dimensional model for aspiration in blood vessels that accounts for the elasticity of both the vessel wall and the catheter. The inclusion of vessel wall viscoelasticity transforms the governing equation for the flow rate into a parabolic form, enabling accurate resolution of the sharp pressure gradients near the catheter tip that a purely hyperbolic formulation cannot capture. A simplified catheter equilibrium approximation is proposed that reproduces the fully elastic catheter model with high accuracy while reducing computation time significantly. The numerical treatment is based on a relaxation of the hyperbolic subsystem that yields a Lax-Friedrichs-type finite volume scheme and facilitates nodal solvers, enabling efficient coupling between catheterized and uncatheterized vessel segments, including bifurcations and the catheter tip. An implicit-explicit splitting strategy ensures that the viscoelastic terms incur only negligible additional computational cost relative to the purely hyperbolic model. The model is validated against three-dimensional CFD simulations and reference data from the literature, including a suction-force-suction-distance analysis. Numerical experiments investigating the role of catheter elasticity and suction force on the hemodynamics are presented, and an uncertainty quantification study demonstrates the suitability of the framework for efficient parameter studies.
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Abstract
A new moving mesh scheme based on the Lagrange–Galerkin method for the approximation of the one-dimensional convection–diffusion equation is studied. The mesh movement is prescribed by a discretized dynamical system for the nodal points. This system is related to the velocity and diffusion coefficient in the convection–diffusion equation such that the nodal points follow the convective flow of the model. It is shown that under a restriction of the time step size the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. Using a piecewise linear approximation, optimal error estimates in the $\ell^\infty(L^2) \cap \ell^2(H_0^1)$ norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange–Galerkin method on a fixed mesh exhibits.
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Abstract
We discuss random hyperbolic conservation laws and introduce a formulation interpreting the stochastic variables as additional spatial dimensions with zero flux. The approach is compared to established non–intrusive approaches for random conservation laws. For the numerical approximation a Runge–Kutta discontinuous Galerkin method is employed and a cellwise integration is used for the approximation of the stochastic moments. By means of grid adaptation the computational effort is reduced in the spatial as well as in the stochastic directions, simultaneously. Results on Burgers’ equation are validated by several numerical examples and compared to Monte Carlo simulations.
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Abstract
We considered the simulation of isentropic flow in pipelines and pipe networks. Standard operating conditions in pipe networks suggested an emphasis to simulate low Mach and high friction regimes—however, the system was stiff in these regimes and conventional explicit approximation techniques proved quite costly and often impractical. To combat these inefficiencies, we developed a novel asymptotic-preserving scheme that was uniformly consistent and stable for all Mach regimes. The proposed method for a single pipeline followed the flux splitting suggested in Haack et al., in which the flux was separated into stiff and non-stiff portions then discretized in time using an implicit-explicit approach. The non-stiff part was advanced in time by an explicit hyperbolic solver; we opted for the second-order central-upwind finite volume scheme. The stiff portion is advanced in time implicitly using an approach based on Rosenbrock-type Runge-Kutta methods, which ultimately reduced this implicit stage to a discretization of a linear elliptic equation. To extend to full pipe networks, the scheme on a single pipeline was paired with coupling conditions defined at pipe-to-pipe intersections to ensure a mathematically well-posed problem. We showed that the coupling conditions remained well-posed at the low Mach/high friction limit—which, when used to define the ghost cells of each pipeline, resulted in a method that was accurate across these intersections in all regimes. The proposed method was tested on several numerical examples and produced accurate, non-oscillatory results with run times independent of the Mach number.
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Netw. Heterog. Media 20 (2025), no. 1, 254–285
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Abstract
In this paper, a novel multiscale consensus-based optimization (CBO) algorithm for solving bi- and tri-level optimization problems is introduced. Existing CBO techniques are generalized by the proposed method through the employment of multiple interacting populations of particles, each of which is used to optimize one level of the problem. These particle populations are evolved through multiscale-in-time dynamics, which are formulated as a singularly perturbed system of stochastic differential equations. Theoretical convergence analysis for the multiscale CBO model to an averaged effective dynamics as the time-scale separation parameter approaches zero is provided. The resulting algorithm is presented for both bi-level and tri-level optimization problems. The effectiveness of the approach in tackling complex multilevel optimization tasks is demonstrated through numerical experiments on various benchmark functions. Additionally, it is shown that the proposed method performs well on min–max optimization problems, comparing favorably with existing CBO algorithms for saddle point problems.
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Math. Models Methods Appl. Sci. 35 (2025), no. 10, 2207–2243
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Abstract
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.
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Abstract
A recently introduced coupling strategy for two nonconservative hyperbolic systems is employed to investigate a collapsing vapor bubble embedded in a liquid near a solid. For this purpose, an elastic solid modeled by a linear system of conservation laws is coupled to the two-phase Baer-Nunziato-type model for isothermal fluids, a nonlinear hyperbolic system with nonconservative products. For the coupling of the two systems the Jin-Xin relaxation concept is employed and embedded in a second order finite volume scheme. Numerical simulations studying the collapsing bubble experiment in one space dimension are presented.
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Applied mathematics and computation 504, Issue C (2025)
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Abstract
Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter $\delta > 0$ in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as $\delta \to 0$. Moreover, we show that any sequence of minimizers subconverges in $C^1$, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter $\delta$ is sufficiently small. Therefore, this limit curve serves as an “optimal” curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.
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Nonlinear Anal. 263 (2026), Paper No. 113942
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Abstract
Motivated by the development of dynamics in probability spaces, we propose a novel multi-agent dynamic of consensus type where each agent is a probability measure. The agents move instantaneously towards a weighted barycenter of the ensemble according to the 2-Wasserstein metric. We mathematically describe the evolution as a system of measure differential inclusions and show the existence of solutions for compactly supported initial data. Inspired by the consensus-based optimization, we apply the multi-agent system to solve a minimization problem over the space of probability measures. In the small numerical example, each agent is described by a particle approximation and aims to approximate a target measure.
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SIAM J. Math. Anal. 57 (2025), no. 5, 5107–5134
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Abstract
We study a linear model for the propagation of acoustic and surface gravity waves in a stratified free-surface ocean. A formulation was previously obtained by linearizing the compressible Euler equations. In this paper, we introduce a new formulation written with a generalized potential. The new formulation is obtained by studying the functional spaces and operators associated to the model. The mathematical study of this new formulation is easier and the discretization is also more efficient than for the previous formulation. We prove that both formulations are well-posed and show that the solution to the first formulation can be obtained from the solution to the second. Finally, the formulations are discretized using a spectral element method, and we simulate tsunamis generation from submarine earthquakes and landslides.
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ESAIM Math. Model. Numer. Anal. 59 (2025), no. 1, 1-41
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Abstract
We address an optimization problem where the cost function is the expectation of a random mapping. To tackle the problem two approaches based on the approximation of the objective function by consensus-based particle optimization methods on the search space are developed. The resulting methods are mathematically analyzed using a mean-field approximation and their connection is established. Several numerical experiments show the validity of the proposed algorithms and investigate their rates of convergence.
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SIAM J. Optim. 35 (2025), no. 4, 2572-2598
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Abstract
The interest in boundary feedback control of multi-dimensional hyperbolic systems is increasing. In the present work we want to compare some of the recent results available in the literature.
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Abstract
Recently the barotropic two fluid model belonging to the class of symmetric hyperbolic thermodynamically compatible (SHTC) systems was studied in detail in Thein, Romenski, and Dumbser (2022). There the question was raised whether the dissipative structure introduced by the source terms satisfies the Shizuta-Kawashima condition. This well-known condition is a sufficient criterion for the existence of global smooth solutions of the studied system. In this work we exploit the dissipative structure of the system under consideration and verify that the Shizuta-Kawashima condition holds.
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Abstract
The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.
Reference
J. Comput. Phys. 518 (2024), 113330