Abstract
A recently introduced scheme for networked conservation laws is analyzed in various experiments. The scheme makes use of a novel relaxation approach that governs the coupling conditions of the network and does not require a solution of the Riemann problem at the nodes. We numerically compare the dynamics of the solution obtained by the scheme to solutions obtained using a classical coupling condition. In particular, we investigate the case of two outgoing edges in the Lighthill–Whitham–Richards model of traffic flow and in the Buckley–Leverett model of two phase flow. Moreover, we numerically study the asymptotic preserving property of the scheme by comparing it to its preliminary form before the relaxation limit in a 1-to-1 network.
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Proc. Appl. Math. Mech., 23: e202200150
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Abstract
This paper is concerned with the Boltzmann equation based on a continuous internal energy variable to model polyatomic gases with constant specific heats. We propose a family of models for the collision kernel and evaluate the nonlinear Boltzmann collision operator to get explicit expressions for transport coefficients like shear and bulk viscosities, thermal conductivity, depending on the collision kernel parameters. This model is shown to contain as a special case the collision kernel used in the direct simulation Monte Carlo method with the variable hard sphere cross section. Then, we show that it is possible to choose parameters in such a way that we recover various physical phenomena, in particular, experimental data for the shear viscosity, Prandtl number and the ratio of bulk and shear viscosities at the same time.
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Abstract
Traffic flow on networks requires knowledge on the behavior across traffic intersections. For macroscopic models based on hyperbolic conservation laws there exist nowadays many ad-hoc models describing this behavior. Based on car trajectory data we propose a novel framework combining data-fitted models with the requirements of consistent coupling conditions for macroscopic models of traffic junctions. A method for deriving density and flux corresponding to the traffic close to the junction for data-driven models is presented. Within the models parameter fitting as well as machine-learning approaches enter to obtain suitable boundary conditions for macroscopic first and second-order traffic flow models. The prediction of various models are compared considering also existing coupling rules at the junction. Numerical results imposing the data-fitted coupling models on a traffic network are presented.
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accepted for Journal Mathematical Methods in the Applied Sciences
Abstract
In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in Romenski et al. (J Sci Comput 42(1):68, 2009, Quart Appl Math 65(2):259–279, 2007). All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine–Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer–Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer–Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer–Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer–Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.
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Abstract
Two phase flows that include phase transition, especially phase creation, with a sharp interface remain a challenging task for numerics. We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The phase interface is modeled as a sharp interface and the mass transfer across the phase boundary is modeled by a kinetic relation. Existence and uniqueness results were proven in Ref. \cite{Hantke2019a}. Using sharp interfaces for simulating nucleation and cavitation results in the grid containing tiny cells that are several orders of magnitude smaller than the remaining grid cells. This forces explicit time stepping schemes to take tiny time steps on these cells. As a remedy we suggest an explicit implicit domain splitting where the majority of the grid cells is treated explicitly and only the neighborhood of the tiny cells is treated implicitly. We use dual time stepping to solve the resulting small implicit systems. Our numerical results indicate that the new scheme is robust and provides significant speed-up compared to a fully explicit treatment.
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Physics of Fluids, Vol. 35, Issue 1
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Abstract
We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure Appl. Math. 48(3), 235-276 (1995)] and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative and yields well defined and easy-to-compute coupling conditions even for general networks. We discuss higher order extension of the scheme and applications to traffic flow and two-phase flow. In the former we compare with results obtained in literature.
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Netw. Heterog. Media 18 (2023), no.1, 310–340
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Abstract
In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated via scalarization using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space, in particular over the Pareto front, by using energy-based measures to quantify the diversity of the system. The resulting gradient-free metaheuristic algorithm is mathematically analyzed using a mean-field approximation of the algorithm iteration and convergence guarantees towards Pareto optimal points are rigorously proven. In addition, we analyze the dynamics when the Pareto front corresponds to the unit simplex, and show that the adaptive mechanism reduces to a gradient flow in this case. Several numerical experiments show the validity of the proposed stochastic particle dynamics, investigate the role of the algorithm parameters and validate the theoretical findings.
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Abstract
We are interested in the feedback stabilization of systems described by Hamilton-Jacobi type equations in $\R^n$. A reformulation leads to a a stabilization problem for a multi-dimensional system of n hyperbolic partial differential equations. Using a novel Lyapunov function taking into account the multi-dimensional geometry we show stabilization in $L^2$ for the arising system using a suitable feedback control. We further present examples of such systems partially based on a forming process.
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Abstract
Aronszajn, Krzywicki and Szarski proved a strong unique continuation result for differential forms, satisfying a certain first order differential inequality, on Riemannian manifolds with empty boundary. The present paper extends this result to the setting of Riemannian manifold with non-empty boundary, assuming suitable boundary conditions on the differential forms. We then present some applications of this extended result. Namely, we show that the Hausdorff dimension of the zero set of harmonic Neumann and Dirichlet forms, as well as eigenfields of the curl operator (on 3-manifolds), has codimension at least 2. Again, these bounds were known in the setting of manifolds without boundary, so that the merit is once more the inclusion of boundary points.
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Abstract
We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O’Hara’s self-repulsive potentials $ E^{\alpha,p}$. In order to do so, we employ the theory of curves of maximal slope in slightly smaller spaces compactly embedding into the respective energy spaces associated to these functionals, and add a term involving the logarithmic strain, which controls the parametrisations of the flowing (knotted) loops. As a prerequisite, we prove in addition that O’Hara’s knot energies $E^{\alpha,p}$ are continuously differentiable.
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Abstract
We present a multi-agent algorithm for multi-objective optimization problems, which extends the class of consensus-based optimization methods and relies on a scalarization strategy. The optimization is achieved by a set of interacting agents exploring the search space and attempting to solve all scalar sub-problems simultaneously. We show that those dynamics are described by a mean-field model, which is suitable for a theoretical analysis of the algorithm convergence. Numerical results show the validity of the proposed method.
Reference
2022 IEEE 61st Conference on Decision and Control (CDC), Cancun, Mexico, 2022, pp. 4131-4136
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Abstract
Using interpolation with biarc curves we prove $\Gamma$-convergence of discretized tangent-point energies to the continuous tangent-point energies in the $C^1$-topology, as well as to the ropelength functional. As a consequence discrete almost minimizing biarc curves converge to ropelength minimizers, and to minimizers of the continuous tangent-point energies. In addition, taking point-tangent data from a given $C^{1,1}$-curve $\gamma$, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of $\gamma$, together with an explicit convergence rate.
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Adv. Contin. Discrete Models 2023, Paper No. 4, 33 pp.
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Abstract
A multiresolution analysis (MRA) for solving stochastic conservation laws is proposed. Using a novel adaptation strategy and a higher-dimensional deterministic problem, a discontinuous Galerkin (DG) solver is derived. An MRA of the DG spaces for the proposed adaptation strategy is presented. Numerical results show that in the case of general stochastic distributions the performance of the DG solver is significantly improved by the novel adaptive strategy. The gain in efficiency is validated in computational experiments.
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IMA Journal of Numerical Analysis, 2023, drad010
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Abstract
We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by removing the connected boundary assumption.
Reference
Math. Anal. Appl. 519 (2023), no. 2, Paper No. 126808