Abstract

An example by Bastin and Coron illustrates that the boundary stabilization of 1-d hyperbolic systems with certain source terms is only possible if the length of the space interval is sufficiently small.

We show that related phenomena also occur for networks of vibrating strings that are governed by the wave equation with a certain source term. It turns out that for a tree of strings with Neumann velocity feedback control at one boundary node and a homogeneous Dirichlet boundary condition at at least one boundary node and homogeneous Dirichlet or Neumann conditions at the other boundary nodes, boundary feedback stabilization is not possible if one of the strings is sufficiently long. However, if the number of strings in the tree is sufficiently large, also for arbitrarily short strings for certain parameters in the source term stabilization is not possible.

The wave equation with source term that we consider is equivalent to a certain 2 ×2 system. For the examples that illustrate the limits of stabilizability, the matrix of the source term is not positive definite. However if the system parameters are chosen in such a way that the matrix is positive semi-definite, the tree of strings can be stabilized exponentially fast by the boundary feedback control for arbitrary long space intervals.

Reference

Systems & Control Letters, Volume 131, 2019

Abstract

Physical systems such as water and gas networks are usually operated in a state of equilibrium and feedback control is employed to damp small perturbations over time. We consider flow problems on networks, described by hyperbolic balance laws, and analyze the stability of the linearized systems. Sufficient conditions for exponential stability in the continuous and discretized setting are presented. The analysis is extended to arbitrary Sobolev norms. Computational experiments illustrate the theoretical findings.

Reference

Mathematical Control & Related Fields, 2019, 9 (3) : 517-539

Abstract

The Riemann problem for coupled Euler equations is analysed. The coupling conditions at a steady interface impose continuous pressure and temperature while momentum differs. The outtake of the momentum models the influence of a gas-powered generator linked to a high-pressure gas network. We prove the existence and uniqueness of the solution to the coupled Riemann problem in case the drop in the momentum is sufficiently small. Furthermore, we analyse the coupling problem for the special case of isentropic Euler equations and obtain similar results. The behaviour of coupled isentropic and coupled compressible Euler equations is compared numerically.

Reference

Vietnam J. Math. 47 (2019), 769–792

Abstract

Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as non-hyperbolic moment systems. Entropy-based Stochastic Galerkin methods, on the other hand, guarantee hyperbolicity and entropy decay. In this paper, we review recent developments of entropy-based stochastic Galerkin methods. A key challenge facing these methods is computational cost, since they require solving a non-linear optimization problem in each time iteration for every spatial cell. Furthermore, the spatial and temporal discretization needs to preserve realizability, meaning that it needs to ensure the existence of a unique solution to the non-linear optimization problem. We review strategies to guarantee realizability, which use a special choice of the numerical flux while considering errors from the optimization solve. Most importantly, we indicate how intrusive entropy-based closures can be made competitive. We show several numerical test cases, among them 2D Euler flow through a nozzle and over an airfoil. Along the way, we discuss the advantages and disadvantages of several uncertainty propagation methods for hyperbolic conservation laws.

Reference

Accepted: Proceedings of Numhyp 2019

Numerical methods for hyperbolic problems 2019 Malaga, 17-21 June 2019

Abstract

The paper is concerned with the numerical solution of the linear steady-state regularized 13-moment equations in two space dimensions. To facilitate the understanding of the specific challenges, the equations are first divided into two subsystems before the full system is approached. The arising problems mainly stem from the complicated saddle-point structure as well as the non-standard nature of the boundary conditions. A continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated. To render the presented approach more efficient, a hybridization technique is presented that originates in the discontinuous Galerkin method.

Reference

J. Comput. Phys. 389 (2019), 1–21

Abstract

Solving the radiation transport equation is a challenging task, due to the high dimensionality of the solution’s phase space. The commonly used discrete ordinates ($S_N$) method suffers from ray effects which result from a break in rotational symmetry from the finite set of directions chosen by $S_N$. The spherical harmonics ($P_N$) equations, on the other hand, preserve rotational symmetry, but can produce negative particle densities. By construction, the discrete ordinates ($S_N$) method ensures non-negative particle densities.

In this paper we present a modified version of the $S_N$ method, the rotated $S_N$ ($rS_N$) method. Compared to $S_N$, we add a rotation and interpolation step for the angular quadrature points and the respective function values after every time step. Thereby, the number of directions on which the solution evolves is effectively increased and ray effects are mitigated. Solution values on rotated ordinates are computed by an interpolation step. Implementation details are provided and in our experiments the rotation and interpolation steps only add 5% to 10% to the runtime of the $S_N$ method. We apply the $rS_N$ method to the line-source and a lattice test case, both being prone to ray effects. Ray effects are reduced significantly, even for small numbers of quadrature points. The $rS_N$ method yields qualitatively similar solutions to the $S_N$ method with less than a third of the number of quadrature points, both for the line-source and the lattice problem. The code used to produce our results is freely available and can be downloaded [4].

Reference

J. Comput. Phys. 382 (2019), 105–123

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arXiv:1808.05846

Abstract

We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We show that this equation still retains many of the properties of the original equations, including hyperbolicity, an entropy-dissipation law, and rotational invariance. The cost of the regularization is mismatch between the moment vector of the solution and that of the ansatz returned by the regularized optimization problem. However, we show how to control this error using the parameter defining the regularization. This suggests that with proper choice of the regularization parameter, the new method can be used to generate accurate solutions of the original entropy-based moment method, and we confirm this with numerical simulations.

Reference

SIAM J. Appl. Math. 79 (2019), no. 5, 1627–1653

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arXiv:1804.05447

Abstract

We present a positive- and asymptotic-preserving numerical scheme for solving linear kinetic transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and realizability limiters in the angular discretization. Under mild assumptions, the scheme becomes a consistent numerical discretization for the limiting diffusion equation when the scattering cross-section tends to infinity. The scheme also preserves positivity of the particle concentration on the space-time mesh and therefore fixes a common defect of spectral angular discretizations. The scheme is tested on the well-known line source benchmark problem with the usual uniform material medium as well as a medium composed of different materials that are arranged in a checkerboard pattern. We also tested the scheme on a Riemann problem with a nonuniform material medium. The observed order of space-time accuracy of the proposed scheme is reported.

Reference

SIAM J. Sci. Comput. 41 (2019), no. 3, A1500–A1526

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arXiv:1807.06109

Abstract

In this paper, we introduce a large system of interacting financial agents in which all agents are faced with the decision of how to allocate their capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model [30]. The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of power-laws in financial data. We follow a kinetic approach similar to [33] and derive the mean field limit of the microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opposing economic concepts of modeling financial agents to be rational or boundedly rational. Furthermore, this is to our knowledge the first kinetic portfolio model which considers a wealth and stock price distribution simultaneously. Due to the kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a log-normal law. For the stock price distribution, we can either observe a log-normal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibit a fat-tail, which is a well known characteristic of real financial data.

Reference

Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 11, 6209–6238

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arXiv:1711.03291

Abstract

Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots to specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.

Reference

SMAI J. Comput. Math. 5 (2019), 23–51

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arXiv:1712.06966

Abstract

The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $\tau > \sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $\tau > \sqrt{6}$ the support is proven to have Hausdorff dimension at most 6/7.

Reference

J. Calc. Var. (2019) 58:205

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arXiv:1808.09754v1

Abstract

Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from deterministic microscopic agent dynamics. The dynamics are given by a particular class of ordinary differential equations, for which an optimal strategy can be computed (Bressan, Milan Journal of Mathematics, 2011). We use the concept of Nash equilibria and apply the dynamic programming principle to derive the mean field limit equations and we study the scaling behavior of the system as the number of agents tends to infinity and find several mean field game limits. Especially we avoid in our derivation the notion of measure derivatives. Novel scales are motivated by an example of an agent-based financial market model.

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arXiv:1910.13534

Abstract

In this paper, we introduce an extension of a splitting method for singularly perturbed equations, the so-called RS-IMEX splitting [Kaiser et al., Journal of Scientific Computing, 70(3), 1390–1407], to deal with the fully compressible Euler equations. The straightforward application of the splitting yields sub-equations that are, due to the occurrence of complex eigenvalues, not hyperbolic. A modification, slightly changing the convective flux, is introduced that overcomes this issue. It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler equations; numerical results using finite volume and discontinuous Galerkin schemes show the potential of the discretization.

Reference

Commun. Comput. Phys., 27 (2020), pp. 292-320

Abstract

In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the given minimisation problem. Further we generalise Arnold’s results concerning the problem of finding the minimum magnetic energy in an orbit of the group of volume-preserving diffeomorphisms to the setting of abstract manifolds with boundary.

Reference

Ann. Global Anal. Geom. 58 (2020), no. 3, 267–285

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arXiv:1907.05727v3

Abstract

We analyze properties of stochastic hyperbolic systems using a Galerkin formulation, which reformulates the stochastic system as a deterministic one that describes the evolution of polynomial chaos modes. We investigate conditions such that the resulting systems are hyperbolic. We state the eigendecompositions in closed form. A Roe flux is presented and theoretical results are illustrated numerically.

Reference

J. Comput. Phys. 395 (2019), 186–204

Abstract

We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface plays an important role. The convergence rates obtained are optimal given the assumptions on the initial data.

Reference

Interfaces Free Bound. 21 (2019), no. 1, 21–40

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arXiv:1709.04833

Abstract

We introduce a variational time discretization for the multidimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each time step requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.

Reference

Trans. Amer. Math. Soc. 371 (2019), 5083-5155

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arXiv:1411.1012