Abstract
A numerical method solving moment equations with a large number of moments in the gas kinetic theory is presented. The distribution function is expanded in series with the product of Laguerre polynomials and spherical harmonics as basis functions, and a special moment closure is applied to achieve global hyperbolicity. The linear collision terms, including the BGK model, the Shakhov model and the linearized hard-sphere model are considered. Numerical results are validated by comparison with the DSMC results, and the differences between various collision models are exhibited.
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Abstract
The paper is concerned with the discretization of a linearized subsystem of the regularized 13-moment equations. These equations state an approximation of the Boltzmann equation as a result of applying moment methods. First, the derivation of the linearized equations is outlined, followed by the introduction of the numerical approach. The subsystem is of elliptic nature which makes finite elements the method of choice. The handling of saddle-point structures within the equations and non-standard boundary conditions are discussed. In this context, the concept of stabilization is presented and applied to the specific problem.
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Abstract
Kinetic equations like the Boltzmann equation are the basis for various applications involving rarefied gases. An important problem of many approaches since the first developments by Grad is the desired global hyperbolicity of the emerging set of partial differential equations. Due to lack of hyperbolicity of Grad’s model equations, numerical computations can break down or yield nonphysical solutions. New hyperbolic PDE systems for the solution of the Boltzmann equation can be derived using quadrature-based projection methods. The method is based on a non-linear transformation of the velocity to obtain a Lagrangian velocity phase space description in order to allow for physical adaptivity, followed by a series expansion of the unknown distribution function in different basis functions and the application of quadrature-based projection methods. In this paper, we extend the proof for global hyperbolicity of the quadrature-based moment system system to arbitrary dimensions, utilizing quadrature-based projection methods for tensor product Hermite basis functions. The analytical computation of the eigenvalues shows the proposed correspondence to the Hermite quadrature points.
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Abstract
We derive hyperbolic PDE systems for the solution of the Boltzmann Equation. First, the velocity is transformed in a non-linear way to obtain a Lagrangian velocity phase space description that allows for physical adaptivity. The unknown distribution function is then approximated by a series of basis functions. Standard continuous projection methods for this approach yield PDE systems for the basis coefficients that are in general not hyperbolic. To overcome this problem, we apply quadrature-based projection methods which modify the structure of the system in the desired way so that we end up with a hyperbolic system of equations. With the help of a new abstract framework, we derive conditions such that the emerging system is hyperbolic and give a proof of hyperbolicity for Hermite ansatz functions in one dimension together with Gauss-Hermite quadrature.
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Abstract
We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum-entropy M${}_n$ models. A realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closure. They provide nonnegative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite difference Fokker-Planck scheme.
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SIAM J. Appl. Math. 74 (2014), no. 4, 1087–1114
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Abstract
Magnets without inversion symmetry are a prime example of a solid-state system featuring topological solitons on the nanoscale, and a promising candidate for novel spintronic applications. Magnetic chiral skyrmions are localized vortex-like structures, which are stabilized by antisymmetric exchange interaction, the so-called Dzyaloshinskii–Moriya interaction. In continuum theories, the corresponding energy contribution is, in contrast to the classical Skyrme mechanism from nuclear physics, of linear gradient dependence and breaks the chiral symmetry. In the simplest possible case of a ferromagnetic energy in the plane, including symmetric and antisymmetric exchange and Zeeman interaction, we show that the least energy in a class of fields with unit topological charge is attained provided the Zeeman field is sufficiently large.
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Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2172, 20140394, 17 pp.
Abstract
We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.
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Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 1019–1043
Abstract
In micromagnetics, the fundamental evolution law for the magnetization $\mathbf{m}$ in a solid is given by the Landau-Lifshitz-Gilbert equation
$$ \frac{\partial \mathbf{m}} {\partial t} = \mathbf{m} \times \left (\alpha \frac{\partial \mathbf{m}} {\partial t} -\gamma ,\boldsymbol{ h}_{\mathrm{eff}}\right ), \qquad (1) $$
which is used to describe the dynamics of a great variety of magnetic microstructures, in particularly the motion of domain walls and vortices in thin films, see e.g. [3]. Here $\boldsymbol{ h}_{\mathrm{eff}}$ is the effective field, essentially the $L^2$ gradient of the micromagnetic energy.
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phenomena and scaling in mathematical models, 113–132, Springer, Cham, 2014
Abstract
The paper is concerned with the convergence analysis of a numerical method for nonlocal Cahn–Hilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed.
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Abstract
We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.
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Abstract
We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Klein’s non-stiff/stiff decomposition of the fluxes [J. Comput. Phys., 121 (1995), pp. 213–237] with an explicit/implicit time discretization [F. Cordier, P. Degond, and A. Kumbaro, J. Comput. Phys., 231 (2012), pp. 5685–5704] for the split fluxes. This results in a scalar second order partial differential equation (PDE) for the pressure, which we solve by an iterative approximation. Due to our choice of a crucial reference pressure, the stiff subsystem is hyperbolic, and the second order PDE for the pressure is elliptic. The scheme is also uniformly asymptotically consistent. Numerical experiments show that the scheme needs to be stabilized for low Mach numbers. Unfortunately, this affects the asymptotic consistency, which becomes nonuniform in the Mach number, and requires an unduly fine grid in the small Mach number limit. On the other hand, the CFL number is only related to the nonstiff characteristic speeds, independently of the Mach number. Our analytical and numerical results stress the importance of further studies of asymptotic stability in the development of asymptotic preserving schemes.
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Abstract
We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system with so-called Dobrushin boundary conditions, i.e., inhomogeneous $\pm 1$ Dirichlet boundary conditions that enforce at least one transition layer from $-1$ to $1$. (Our methods can be applied to other boundary conditions as well.) We are interested in the competition between the “energy” that should be minimized due to the small noise strength and the “entropy” that is induced by the large system size. Specifically, in the context of system sizes that are exponential with respect to the inverse noise strength — up to the “critical” exponential size predicted by the heuristics — we study the extremely strained large deviation event of seeing more than the one transition layer between $\pm 1$ that is forced by the boundary conditions. We capture the competition between energy and entropy through upper and lower bounds on the probability of these unlikely extra transition layers. Our bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from $-1$ to $+1$ is exponentially close to one. Our second result then studies the distribution of the transition layer. In particular, we establish that, on a super-logarithmic scale, the position of the transition layer is approximately uniformly distributed. In our arguments we use local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.
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Abstract
The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books.