Abstract

We are interested in the numerical solution of large systems of hyperbolic conservation laws or systems in which the characteristic decomposition is expensive to compute. Solving such equations using finite volumes or discontinuous Galerkin requires a numerical flux function which solves local Riemann problems at cell interfaces. There are various methods to express the numerical flux function. On the one end, there is the robust but very diffusive Lax–Friedrichs solver; on the other end is the upwind Godunov solver which respects all resulting waves. The drawback of the latter method is the costly computation of the eigensystem. This work presents a family of simple first-order Riemann solvers, named, HLLX$\omega$, which avoid solving the eigensystem. The new method reproduces all waves of the system with less dissipation than other solvers with similar input and effort, such as HLL and FORCE. The family of Riemann solvers can be seen as an extension or generalization of the methods introduced by Degond et al. [C.R. Acad. Sci. Paris Ser. I, 328 (1999), pp. 479–483]. We only require the same number of input values as HLL, namely, the globally fastest wave speeds in both directions, or an estimate of the speeds. Thus, the new family of Riemann solvers is particularly efficient for large systems of conservation laws when the spectral decomposition is expensive to compute or no explicit expression for the eigensystem is available.

Reference

SIAM J. Sci. Comput. 39 (2017), no. 6, A2911–A2934

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arXiv:1607.05721

Abstract

A hierarchical simulation approach for Boltzmann’s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains.

The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.

Reference

J. Comput. Phys. 342 (2017), 66–84

Abstract

We present efficient algorithms and implementations of the 35-moment system equipped with the maximum-entropy closure in the context of rarefied gases. While closures based on the principle of entropy maximization have been shown to yield very promising results for moderately rarefied gas flows, the computational cost of these closures is in general much higher than for closure theories with explicit closed-form expressions of the closing fluxes, such as Grad’s classical closure. Following a similar approach as Garrett et al. (2015) [13], we investigate efficient implementations of the computationally expensive numerical quadrature method used for the moment evaluations of the maximum-entropy distribution by exploiting its inherent fine-grained parallelism with the parallelism offered by multi-core processors and graphics cards. We show that using a single graphics card as an accelerator allows speed-ups of two orders of magnitude when compared to a serial CPU implementation. To accelerate the time-to-solution for steady-state problems, we propose a new semi-implicit time discretization scheme. The resulting nonlinear system of equations is solved with a Newton type method in the Lagrange multipliers of the dual optimization problem in order to reduce the computational cost. Additionally, fully explicit time-stepping schemes of first and second order accuracy are presented. We investigate the accuracy and efficiency of the numerical schemes for several numerical test cases, including a steady-state shock-structure problem.

Reference

J. Comput. Phys. 340 (2017), 138–159

Abstract

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem.

Reference

Eur. J. Mech. B Fluids 64 (2017), 30–40

Abstract

The Boltzmann equation can be used to model rarefied gas flows in the transition or kinetic regime, i.e. for moderate to large Knudsen numbers. However, standard moment methods like Grad’s approach lack hyperbolicity of the equations. This can lead to instabilities and nonphysical solutions. Based on recent developments in this field we have recently derived a quadrature-based moment method leading to globally hyperbolic and rotationally invariant moment equations. We present a 1D five moment case of the equations and use numerical simulations to compare the new model with standard approaches. The tests are done with dedicated numerical methods to solve the new non-conservative moment equations. These first results using the new method show the accuracy of the new method and its benefits compared with Grad’s method or other existing models like discrete velocity.

Reference

Eur. J. Mech. B Fluids 64 (2017), 41–46

Abstract

Moment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad’s equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.

In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

Reference

Commun. Comput. Phys. 21 (2017), no. 4, 981–1011

Abstract

Particle transport in radiation therapy can be modelled by a kinetic equation which must be solved numerically. Unfortunately, the numerical solution of such equations is generally too expensive for applications in medical centers. Moment methods provide a hierarchy of models used to reduce the numerical cost of these simulations while preserving basic properties of the solutions. Moment models require a closure because they have more unknowns than equations. The entropy-based closure is based on the physical description of the particle interactions and provides desirable properties. However, computing this closure is expensive. We propose an approximation of the closure for the first two models in the hierarchy, the $M_1$ and $M_2$ models valid in one, two or three dimensions of space. Compared to other approximate closures, our method works in multiple dimensions. We obtain the approximation by a careful study of the domain of realizability and by invariance properties of the entropy minimizer. The $M_2$ model is shown to provide significantly better accuracy than the $M_1$ model for the numerical simulation of a dose computation in radiotherapy. We propose a numerical solver using those approximated closures. Numerical experiments in dose computation test cases show that the new method is more efficient compared to numerical solution of the minimum entropy problem using standard software tools.

Reference

J. Sci. Comput. 71 (2017), no. 1, 71-108

Abstract

In this paper, we study the sensitivities of electron dose calculations with respect to stopping power and transport coefficients. We focus on the application to radiotherapy simulations. We use a Fokker-Planck approximation to the Boltzmann transport equation. Equations for the sensitivities are derived by the adjoint method. The Fokker-Planck equation and its adjoint are solved numerically in slab geometry using the spherical harmonics expansion $(P_N)$ and an Harten-Lax-van Leer finite volume method. Our method is verified by comparison to finite difference approximations of the sensitivities. Finally, we present numerical results of the sensitivities for the normalized average dose deposition depth with respect to the stopping power and the transport coefficients, demonstrating the increase in relative sensitivities as beam energy decreases. This in turn gives estimates on the uncertainty in the normalized average deposition depth, which we present.

Reference

Math. Med. Biol. 34 (2017), no. 1, 109-123

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arXiv:1410.5622

Abstract

The Landau–Lifshitz–Gilbert–Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin-polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space–time patterns in the form of wavetrains and coherent structures, whose local wavenumber varies in space. A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetrains or the constant up-/down-magnetizations. For certain polarization, the Slonczewski term can be removed which allows for a more complete characterization, including soliton-type solutions. Decisive for the solution structure is the polarization parameter as well as size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping.

Reference

J. Nonlinear Sci. 27 (2017), no. 5, 1551-1587

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arXiv:1309.5523

Abstract

We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The singular set of the solution coincides with the points at which the delta measures are centered.

Reference

Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017), no. 1, 113-143

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arXiv:1601.01926

Abstract

A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29–58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050–2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro Díaz, and C. Parés, Appl. Math. Comput., 219 (2013), pp. 9012–9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales' scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

Reference

SIAM J. Numer. Anal. 55 (2017), no. 2, 758-784

Abstract

In this work, we propose a new way of splitting the flux function of the isentropic compressible Euler equations at low Mach number into stiff and non-stiff parts. Following the IMEX methodology, the latter ones are treated explicitly, while the first ones are treated implicitly. The splitting is based on the incompressible limit solution, which we call reference solution. An analysis concerning the asymptotic consistency and numerical results demonstrate the advantages of this splitting.

Reference

J. Sci. Comput. 70 (2017), no. 3, 1390-1407

Abstract

We investigate a connection between the two-dimensional Finslerian area functional in arbitrary codimension based on the Busemann–Hausdorff volume form, and well-investigated Cartan functionals to solve the Plateau problem in Finsler spaces. This generalises a previously known result due to Overath and von der Mosel (Manuscripta Math 143(3–4):273–316, 2014) to higher codimension.

Reference

Eur. J. Math. 3 (2017), no. 4, 953-973

Abstract

To describe the behavior of knotted loops of springy wire with an elementary mathematical model we minimize the integral of squared curvature, $E= \int \varkappa^2$, together with a small multiple of ropelength $R$ = length/thickness in order to penalize selfintersection. Our main objective is to characterize all limit configurations of energy minimizers of the total energy $E_\vartheta \equiv E+\vartheta R$ as $ϑ$ tends to zero. For short, these limit configurations will be referred to as _elastic knots_. The elastic unknot turns out to be the once covered circle with squared curvature energy $(2π)^2$. For all (non-trivial) knot classes for which the natural lower bound $(4π)^2$ on $E$ is sharp, the respective elastic knot is the doubly covered circle. We also derive a new characterization of two-bridge torus knots in terms of $E$, proving that the only knot classes for which the lower bound $(4π)^2$ on $E$ is sharp are the $(2,b)$-torus knots for odd $b$ with $|b| \ge 3$ (containing the trefoil knot class). In particular, the elastic trefoil knot is the doubly covered circle.

Reference

Arch. Ration. Mech. Anal. 225 (2017), no. 1, 89-139

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arxiv:1510.06171

Abstract

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a “droplet-shaped” local energy minimizer are established. A standard mountain pass argument leads to the existence of a saddle point whose energy is equal to the energy barrier, for which a quantitative bound is deduced. In addition, finer properties of the local minimizer and appropriately defined constrained minimizers are deduced. The proofs employ the $\Gamma$-limit (identified in a previous work), quantitative isoperimetric inequalities, and variational arguments.

Reference

Indiana Univ. Math. J. 66 (2017), no. 5, 1827-1877