Abstract
Motivated by recent models of current driven magnetization dynamics, we examine the coupling of the Landau-Lifshitz-Gilbert equation and classical electron transport governed by the Vlasov-Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields of quantized helicity that add up to the conventional Maxwell fields. We construct global weak solutions of the coupled system in the framework of frustrated magnets with competing first and second order gradient interactions known to host topological solitons such as magnetic skyrmions and hopfions.
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Abstract
In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings the class of CBO methods. Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. Using a mean-field description of the the many particle limit of the arising CBO dynamics, we are able to show convergence of the proposed method to the minimum for general nonlinear constrained problems. Properties of the new algorithm are analyzed. Several numerical examples, also in high dimensions, illustrate the theoretical findings and the good performance of the new numerical method.
Reference
SIAM J. Optim. 33 (2023), no. 1, 211–236
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Abstract
In recent years the concept of multiresolution-based adaptive discontinuous Galerkin (DG) schemes for hyperbolic conservation laws has been developed. The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defined on a hierarchy of nested grids. Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element. For non-uniform grid hierarchies multiwavelets have to be constructed for each element and, thus, becomes extremely expensive. To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.
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Abstract
Aiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the $W^{3/2,2}$-inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on $L^2$-gradients.
Reference
Arch. Rational Mech. Anal (Sept. 2021)
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Abstract
Based on a continuous internal energy state variable, we propose an explicit, fully nonlinear Boltzmann collision operator for the evolution of the distribution function describing a polyatomic gas with a constant heat capacity. The particle interaction is a polyatomic generalization of the variable hard-sphere model, used in a recent rigorous mathematical analysis, and includes frozen collisions. The model is consistent with the monatomic case and allows easy evaluations for moment equations and the Chapman-Enskog expansion. Using a publicly available computer algebra code we can explicitly compute nonlinear production terms for macroscopic systems of moments. The range of Prandtl number values recovers the Eucken formula for a specific choice of frozen collisions.
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Abstract
In this paper, we consider the kinetic model of continuous type describing a polyatomic gas in two different settings corresponding to a different choice of the functional space used to define macroscopic quantities. Such a model introduces a single continuous variable supposed to capture all the phenomena related to the more complex structure of a polyatomic molecule. In particular, we provide a direct comparison of these two settings, and show their equivalence after the distribution function is rescaled and the cross section is reformulated. We then focus on the kinetic model for which the rigorous existence and uniqueness result in the space homogeneous case is recently proven. Using the cross section proposed in that analysis together with the maximum entropy principle, we establish macroscopic models of six and fourteen fields. In the case of six moments, we calculate the exact, nonlinear, production term and prove its total agreement with extended thermodynamics. Moreover, for the fourteen moments model, we provide new expressions for relaxation times and transport coefficients in a linearized setting, that yield both matching with the experimental data for dependence of the shear viscosity upon temperature and a satisfactory agreement with the theoretical value of the Prandtl number.
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Abstract
We design a Fortin operator for the lowest-order Taylor-Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $P_2 - P_0$ and the augmented Taylor-Hood element in 3D.
Reference
SIAM J. Numer. Anal. 59 (2021), no. 5, 2571–2607
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Abstract
We propose a new second-order accurate hydrostatic reconstruction scheme for the Saint-Venant system. Such a scheme needs to overcome several difficulties: besides the well-known issues of positivity and well-balancing there is also the difficulty of unphysical reflections from bottom reconstructions which create artificial steps. We address all of these problems at once by changing the logic of the reconstruction of the bottom, the water depth and the water surface level. Notably, our bottom reconstruction is continuous across cell interfaces and remains unchanged during the computation, except if the original topography has a jump, or if a wet-dry front passes through a cell. Only in these exceptional cases we apply the new discontinuous bottom approximation and compute the residual via the subcell hydrostatic reconstruction method. The scheme gives excellent results in one and two space dimensions. To highlight the novel reconstruction of bottom and water surface, we call the scheme bottom-surface-gradient method (BSGM).
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Abstract
In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. For the latter methods, convergence to the global minimizer can be shown for a large class of functions. Algorithmic implementations inspired by the well-known direct simulation Monte Carlo methods in kinetic theory are derived and discussed. Several examples on prototype test functions for global optimization are reported including applications to machine learning.
Reference
Appl. Math. Optim. 86 (2022), no. 1, Paper No. 9
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Abstract
This work is devoted to the structure of the time-discrete Green–Naghdi equations including bathymetry. We use the projection structure of the equations to characterize homogeneous and inhomogeneous boundary conditions for which the semidiscrete equations are well posed. This structure allows us to propose efficient and robust numerical treatment of the boundary conditions that ensures entropy stability of the scheme by construction. Numerical evidence is provided to illustrate that our approach is suitable for situations of practical interest that are not covered by existing theory.
Reference
SIAM J. Numer. Anal. 60 (2022), no. 5, 2681–2712
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Abstract
We discuss random hyperbolic conservation laws and introduce a novel formulation interpreting the stochastic variables as additional spatial dimensions with zero flux. The approach is compared with established non-intrusive approaches to random conservation laws. In the scalar case, an entropy solution is proven to exist if and only if a random entropy solution for the original problem exists. Furthermore, existence and numerical convergence of stochastic moments is established. Along with this, the boundedness of the $L^1$-error of the stochastic moments by the $L^1$-error of the approximation is proven. For the numerical approximation a Runge-Kutta discontinuous Galerkin method is employed and a multi-element stochastic collocation 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 Burger’s and Euler equation are validated by several numerical examples and compared to Monte Carlo simulations.
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Abstract
A maximum entropy dissipation problem at a traffic junction and the corresponding coupling condition are studied. We prove that this problem is equivalent to a coupling condition introduced by Holden and Risebro. An $L^1$-contraction property of the coupling condition and uniqueness of solutions to the Cauchy problem are proved. Existence is obtained by a kinetic approximation of Bhatnagar–Gross–Krook type together with a kinetic coupling condition obtained by a kinetic maximum entropy dissipation problem. The arguments do not require total variation bounds on the initial data compared to previous results. We also discuss the role of the entropies involved in the macroscopic coupling condition at the traffic junction by studying an example.
Reference
SIAM J. Math. Anal. 54 (2022), no. 1, 954–985
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Abstract
In this paper we derive optimal relaxation rates for the Cahn-Hilliard equation on the one-dimensional torus and the line. We consider initial conditions with a finite (but not small) $L^1$-distance to an appropriately defined bump. The result extends the relaxation method developed previously for a single transition layer (the “kink”) to the case of two transition layers (the “bump”). As in the previous work, the tools include Nash-type inequalities, duality arguments, and Schauder estimates. For both the kink and the bump, the energy gap is translation invariant and its decay alone cannot specify to which member of the family of minimizers the solution converges. Whereas in the case of the kink, the conserved quantity singles out the longtime limit, in the case of a bump, a new argument is needed. On the torus, we quantify the (initially algebraic and ultimately exponential) convergence to the bump that is the longtime limit; on the line, the bump-like states are merely metastable and we quantify the initial algebraic relaxation behavior.
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Accepted at Communications in Partial Differential Equations
https://doi.org/10.1080/03605302.2021.1987458
Abstract
Fluid flow in pipes with discontinuous cross section or with kinks is described through balance laws with a non conservative product in the source. At jump discontinuities in the pipes’ geometry, the physics of the problem suggests how to single out a solution. On this basis, we present a definition of solution for a general BV geometry and prove an existence result, consistent with a limiting procedure from piecewise constant geometries. In the case of a smoothly curved pipe we thus justify the appearance of the curvature in the source term of the linear momentum equation. These results are obtained as consequences of a general existence result devoted to abstract balance laws with non conservative source terms in the non resonant case.
Reference
Nonlinear Anal. Real World Appl. 66 (2022), Paper No. 103539, 27 pp.
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Abstract
We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case, and provide $C^{1,1}$-bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical $L^2$ gradient descent or other optimization methods.
Reference
Advances in Calculus of Variations 2022
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Abstract
The modelling of gas networks requires the development of coupling techniques at junctions. Recent work on the coupling of hyperbolic systems based on solving two half Riemann problems can be useful also for the coupling issue in gas networks. This strategy is exemplified here for the coupling of a fluid with a solid modelled by the Euler equations supplemented with a stiffened gas equation and a linear elastic model, respectively. This framework may serve as a basis for investigations of coupling conditions on nodes of a gas network.
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Abstract
We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\mathrm{intM}^{(p,2)}$, with $p \in (\frac{7}{3},\frac{8}{3})$, subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical $C^1$-curves $\gamma : \mathbb{R} / \mathbb{Z} \rightarrow \mathbb{R}^n$ of generalized integral Menger curvature $\mathrm{intM}^{(p,2)}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is inspired by analyticity results on critical points for O’Hara’s knot energies based on Cauchy’s method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.
Reference
Nonlinear Anal. 221 (2022), Paper No. 112858