Abstract

To model shallow free surface flows, the Saint-Venant Equations (SVE) are a convenient simplification of the incompressible Navier–Stokes Equations (NSE). In the present study, we compare the two models for one-dimensional channel flow over a hump (cf. Behr (XNS simulation program, 2016 [5]), Küsters (Comparison of a Navier–Stokes and a shallow water model using the example of flow over a semi-circular bump, 2013 [8]), Noelle et al. (J Comput Phys 226(1):29–58, 2007 [10]), Sikstel (Comparison of hydrostatic and non-hydrostatic shallow water models, 2016 [13])). Our numerical experiments show that the SVE fail for some rather standard transcritical flows, where the two models compute different water heights ahead of and different shock speeds behind the hump. Using numerical computations as well as a formal Cauchy–Kowalevski argument, we give a qualitative explanation of the shortcoming of the SVE. In addition, we examine a recently developed non-hydrostatic shallow water model Sainte-Marie et al. (Discrete and Cont Dyn Syst Ser B 20(4):361–388, 2014 [12]) which proposes to produce physically more realistic results.

Reference

Theory, Numerics and Applications of Hyperbolic Problems II, Springer Proc. Math. Stat., vol 237, Springer, Cham, 2018, pp. 605-616

Abstract

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Reference

Entropy 2018, 20(9), 680

Abstract

To obtain a symmetric hyperbolic moment system from the linearised Boltzmann equation, we approximate the entropy variable (derivative of the entropy functional) with the help of multi-variate polynomials in the velocity space. Choosing the entropy functional to be quadratic, we retrieve the Grad’s approximation for the linearised Boltzmann equation.

We develop a necessary and sufficient condition for the entropy stability of the Grad’s approximation on bounded position domain with inflow and outflow boundaries. These conditions show the importance of using the Onsager Boundary Conditions (OBCs) Rana and Struchtrup (2016) [28] for obtaining entropy stability and we use them to prove that a broad class of Grad’s approximations, equipped with boundary conditions obtained through continuity of odd fluxes Grad (1949) [17], are entropy unstable. Entropy stability, for the Grad’s approximation, is obtained through entropy stabilization of the boundary conditions obtained through the continuity of odd fluxes. Since many practical implementations require the prescription of an inflow velocity, the entropy bounds for two possible methods to achieve the same is discussed in detail, both for the linearised Boltzmann equation and its Hermite approximation. We use the Discontinuous Galerkin (DG) discretization in the physical space, to study several benchmark problems to ascertain the physical accuracy of the proposed entropy stable Grad’s approximation.

Reference

J. Comput. Phys. 369 (2018), 16–44

Abstract

The recent development of the quadrature-based moment equations (QBME) for the simulation of rarefied gases using the Boltzmann equation led to a promising hyperbolic moment model. This paper deals with the two-dimensional QBME model, presents its derivation based on different possible expansions, and gives explicit equations for the QBME model and related models before describing a numerical method to solve the nonconservative PDE system on two-dimensional, unstructured grids. The first simulations using the two-dimensional QBME are shown in this paper using the flow past a cylinder and a forward facing step test case. The results indicate the applicability of the QBME models for rarefied gas flows as well as convergence to the Euler solution in the case of vanishing Knudsen number.

Reference

Multiscale Model. Simul. 16 (2018), no. 2, 1059–1084

Abstract

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

Reference

J. Fluid Mech. 836 (2018), 451–501

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arXiv:1701.09052

Abstract

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.

Reference

J. Stat. Phys. 170 (2018), no. 1, 101–126

Abstract

Hyperbolic moment equations based on Burnett’s expansion of the distribution function are derived for the Boltzmann equation with linearized collision operator. Boundary conditions are equipped for these models, and it is proven that the number of boundary conditions is correct for a large class of moment models. A new second-order numerical scheme is proposed for solving these moment equations, and the new method is suitable for both ordered- and full-moment theories. Numerical experiments are carried out for both one- and two-dimensional problems to show the performance of the moment methods.

Reference

J. Sci. Comput. 74 (2018), no. 1, 336–374

Abstract

Hyperbolicity is a necessary property of model equations for the solution of the BGK equation to achieve stable and physical solutions. However, the standard approach for velocity space discretization developed by Grad only yields locally hyperbolic equations. The method has recently been improved, and several new globally hyperbolic model systems have been derived such as the Hyperbolic Moment Equations (HME) and the Quadrature-Based Moment Equations (QBME). We will describe the derivation and properties of a new model system called Simplified Hyperbolic Moment Equations (SHME) which inherits hyperbolicity from the other models but is simpler to implement and to solve. First simulation results show a good accuracy of the new SHME model in comparison with the existing models.

Reference

Theory, numerics and applications of hyperbolic problems. II, 221–232, Springer Proc. Math. Stat., 237, Springer, Cham, 2018

Abstract

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.

Reference

Kinet. Relat. Models 11 (2018), no. 6, 1503–1526

Abstract

We present results on existence and positivity of solutions for a linear Boltzmann transport equation used for example in radiotherapy applications and more generally in charged particle transports. Therein, some differential cross-sections, that is, kernels of collision integral operators, may become hyper-singular. These collision operators need to be approximated for analytical and numerical treatments. Here, we present an approximation leading to pseudo-differential operators. The final approximation, for which the existence and positivity of solutions is shown, is an integro-partial differential operator which is known as Continuous Slowing Down Approximation (CSDA).

Reference

Math. Models Methods Appl. Sci. 28 (2018), no. 14, 2905–2939

Abstract

In the past decade there has been a growing interest in agent-based econophysical financial market models. The goal of these models is to gain further insights into stylized facts of financial data. We derive the mean field limit of the econophysical Cross model (Cross, 2005) and show that the kinetic limit is a good approximation of the original model. Our kinetic model is able to replicate some of the most prominent stylized facts, namely fat-tails of asset returns, uncorrelated stock price returns and volatility clustering. Interestingly, psychological misperceptions of investors can be accounted to be the origin of the appearance of stylized facts. The mesoscopic model allows us to study the model analytically. We derive steady state solutions and entropy bounds of the deterministic skeleton. These first analytical results already guide us to explanations for the complex dynamics of the model.

Reference

Physica A 505 (2018) 613-631

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arXiv:1711.02573

Abstract

The $M_1$ minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the $M_1$ model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.

Reference

J. Comput. Appl. Math. 342 (2018), 399–418

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arXiv:1706.10174

Abstract

The paper considers a linear Boltzmann transport equation (BTE), and its Continuous Slowing Down Approximation (CSDA). These equations are used to model the transport of particles e.g. in dose calculation of radiation therapy. We prove the existence and uniqueness of weak solutions, under sufficient criteria and in appropriate $L^2$-based spaces, of a single (particle) CSDA-equation by using the theory of $m$-dissipative operators. Relevant a priori estimates are shown as well. In addition, we prove the corresponding results and estimates for a system of coupled transport equations. We also outline a related inverse problem.

Reference

J. Math. Anal. Appl. 460 (2018), no. 1, 271–301

Abstract

We examine a stochastic Landau–Lifshitz–Gilbert equation based on an exchange energy functional containing second-order derivatives of the unknown field. Such regularizations are featured in advanced micromagnetic models recently introduced in connection with nanoscale topological solitons. We show that, in contrast to the classical stochastic Landau–Lifshitz–Gilbert equation based on the Dirichlet energy alone, the regularized equation is solvable in the stochastically strong sense. As a consequence it preserves the topology of the initial data, almost surely.

Reference

IMA J. Appl. Math. 83 (2018), no. 2, 261–282

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arXiv:1705.10184

Abstract

The aim of this thesis is to explore gradient flows in the spaces of currents in general and Cartesian currents (as introduced by Giaquinta, Modica and Soucek) as an important special case. In both cases the focus is on currents without a boundary, however results can be easily extended to the general case. This thesis is split into two parts. The first part is devoted to finding a right choice of a metric for gradient flows, which turns out to be a generalized version of the Wasserstein-distance. The main idea here is to use an analogon of the Benamou-Brenier formulations of optimal transport. To achieve this, first of all, we define a proper notion of transport for currents, in the form of a Lie-derivative. Instead of the conservation of mass, this will lead to a conservation of multiplicity. We then proceed to define two families length-functionals for curves of currents and the resulting generalized Wasserstein-distances, one for the general case and one specifically for Cartesian currents, both parametrized by the exponent $p$. In the case of $p=1$, we show that this generalized distance is in fact equivalent to the flat-metric for currents. For this we use the helpful concept that a curve of $k$-currents of finite $1$-length has a well defined trace in the form of a $k+1$-current. We also show that for Cartesian currents, the newly defined distance results in a form of geometric $L^p$-distance. We then end this part with a discussion of the problem of lower semicontinuity and the correct way to extend previous results to the case of currents with boundary.The second part then details a possible approach for gradient flows of currents using the minimizing movement scheme. Here we show that the powerful compactness properties of currents not only guarantee convergence of the scheme under very mild coercivity and semicontinuity assumptions, but also result in a limit in form of a space-time current. We also discuss various possible improvements to the scheme, as well as connections with the generalized Wasserstein distance. Finally we remark on some possible applications.

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DOI: 10.18154/RWTH-2018-228848

Abstract

We analyze the stability of implicit-explicit flux-splitting schemes for stiff systems of conservation laws. In particular, we study the modified equation of the corresponding linearized systems. We first prove that symmetric splittings are stable, uniformly in the singular parameter ε. Then, we study non-symmetric splittings. We prove that for the isentropic Euler equations, the Degond–Tang splitting [Degond & Tang, Comm. Comp. Phys., 10:1–31, 2011] and the Haack–Jin–Liu splitting [Haack, Jin Liu, Comm. Comp. Phys., 12:955–980, 2012], and for the shallow water equations the recent RS-IMEX splitting are strictly stable in the sense of Majda–Pego. For the full Euler equations, we find a small instability region for a flux splitting introduced by Klein [Klein, J. Comp. Phys., 121:213–237, 1995], if this splitting is combined with an IMEX scheme as in [Noelle, Bispen, Arun, Lukáčová, Munz, SIAM J. Sci. Comp., 36:B989–B1024, 2014].

Reference

Commun. Math. Sci. 16 (2018), no. 1, 1–15

Abstract

Modeling of cavitation phenomena requires the coupling of models for fluid and solid materials. For this purpose we employ a strategy based on the solution of coupled Riemann problems that has been originally developed for the coupling of two fluids. The coupling strategy has been established and validated in [1]. In this work we include a timestepping algorithm which allows for different timesteps in fluid and solid solvers. Furthermore, we perform numerical experiments simulating the interface between a plastic or steel structure and air.

Reference

Proceedings, 6th European Conference on Computational Mechanics / 7th European Conference on Computational Fluid Mechanics, Glasgow, 11-15 June 2018, p. 3049-3059, ISBN 978-84-947311-6-7

Abstract

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in $\mathbb{R}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. The second one is the lower semicontinuity — with respect to Hausdorff-convergence of submanifolds — of all geometric curvature energies under consideration, which can be used to minimise each of these energies within prescribed isotopy classes.

Reference

Comm. Anal. Geom. 26 (2018), no. 6, 1251–1316

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arXiv:1504.04538

Abstract

We establish metastability of the one-dimensional Cahn–Hilliard equation for initial data that is order-one in energy and order-one in $\dot H^{-1}$ away from a point on the so-called slow manifold with $N$ well-separated layers. Specifically, we show that, for such initial data on a system of lengthscale $\Lambda$, there are three phases of evolution: (1) the solution is drawn after a time of order $\Lambda^{2}$ into an algebraically small neighborhood of the $N$-layer branch of the slow manifold, (2) the solution is drawn after a time of order $\Lambda^{3}$ into an exponentially small neighborhood of the $N$-layer branch of the slow manifold, (3) the solution is trapped for an exponentially long time exponentially close to the $N$-layer branch of the slow manifold. The timescale in phase (3) is obtained with the sharp constant in the exponential.

Reference

J. Differential Equations 265 (2018), no. 4, 1528–1575

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arXiv:1705.10985

Abstract

This survey focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces or more general $k$-dimensional sets in $\mathbb{R}^d$. Previous investigations of the authors and collaborators concentrated on the regularising effects of such energies, with a priori estimates in the regime above scale-invariance that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. We briefly describe the impact of geometric curvature energies on geometric knot theory. Currently, various attempts are being made to obtain a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Moreover, a regularity theory for critical points is being developed in the setting of fractional Sobolev spaces. We describe some of these current trends and present a list of open problems.

Reference

In: P. Reiter, S. Blatt, and A. Schikorra (Eds.). New Directions in Geometric and Applied Knot Theory. Berlin, Boston: De Gruyter (pp. 8–35).

Abstract

Gas flow through pipeline networks can be described using $2\times 2$ hyperbolic balance laws along with coupling conditions at nodes. The numerical solution at steady state is highly sensitive to these coupling conditions and also to the balance between flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan[11] introduced a well-balanced method for general $2\times 2$ systems of balance laws. In this paper, we simplify and extend this approach to a network of pipes. We prove well-balancing for different coupling conditions and for compressors stations, and demonstrate the advantage of the scheme by numerical experiments.

Reference

Networks & Heterogeneous Media, 2019, 14 (4) : 659-676

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https://www.igpm.rwth-aachen.de/forschung/preprints/480

Abstract

We consider a hybrid compressible/incompressible system with memory effects introduced by Lefebvre Lepot and Maury (2011) for the description of one-dimensional granular flows. We prove a first global existence result for this system without additional viscous dissipation. Our approach extends the one by Cavalletti, Sedjro, Westdickenberg (2015) for the pressureless Euler system to the constraint granular case with memory effects. We construct Lagrangian solutions based on an explicit formula of the monotone rearrangement associated to the density and explain how the memory effects are linked to the external constraints imposed on the flow. This result is finally extended to a heterogeneous maximal density constraint depending on time and space.

Reference

SIAM J. Math. Anal. 50 (2018), 5921–5946

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arXiv:1703.05829

Abstract

We prove the existence of symmetric critical torus knots for O’Hara’s knot energy family $E_\alpha$, $\alpha \in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth $E_\alpha$-critical knots, which supports experimental observations using numerical gradient flows.

Reference

Topology Appl. 242 (2018), 73–102

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arXiv:1709.06949

In this paper we derive optimal algebraic-in-time relaxation rates to the kink for the Cahn-Hilliard equation on the line. We assume that the initial data have a finite distance - in terms of either a first moment or the excess mass - to a kink profile and capture the decay rate of the energy and the perturbation. Our tools include Nash-type inequalities, duality arguments, and Schauder estimates.

Reference

Accepted: SIAM J. Math. Anal.

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arXiv:1806.02519v2