A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry

Schneider, Florian; Kall, Jochen; Alldredge, Graham W.

doi:10.3934/krm.2016.9.193

Cited by 24 publications

(44 citation statements)

References 38 publications

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“…The mechanism that causes the classical stochastic Galerkin method to loose hyperbolicity has been observed before in the context of high-order discontinuous-Galerkin schemes for hyperbolic systems, especially for moment systems (see, e.g., [3,9,26,29,32,37,38]). We use a similar technique, a "slope limiter", to "dampen" the Gibbs oscillations in the stochastic expansion in such a way that the resulting system is always hyperbolic.…”

Section: Introductionmentioning

confidence: 76%

A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations

Schlachter

Schneider

2018

Journal of Computational Physics

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Uncertainty Quantification through stochastic spectral methods is rising in popularity. We derive a modification of the classical stochastic Galerkin method, that ensures the hyperbolicity of the underlying hyperbolic system of partial differential equations. The modification is done using a suitable "slope" limiter, based on similar ideas in the context of kinetic moment models. We apply the resulting modified stochastic Galerkin method to the compressible Euler equations and the M 1 model of radiative transfer. Our numerical results show that it can compete with other UQ methods like the intrusive polynomial moment method while being computationally inexpensive and easy to implement.

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Section: Introductionmentioning

confidence: 76%

A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations

Schlachter

Schneider

2018

Journal of Computational Physics

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“…Since we need to be able to solve the moment problem to evaluate the right hand side of (3.2), it is crucial to maintain realizability throughout the computation. Similar problems have been treated in the context of radiative transfer, e.g., in [6,16,41,46,48].…”

Section: Realizability-preserving Property Of the Schemesmentioning

confidence: 92%

“…Assuming that γ k i,j is realizable and thatf is the corresponding distribution function approximated by the maximum entropy model. Then to derive a realizable scheme we multiply equation (4.3) with the basis m and integrate over the whole volume domain [25,48], which gives but with respect to the approximate distribution functionf . For the polynomial closure P N we can also use this scheme, but there no realizability can be guaranteed, since the distribution function can get negative.…”

Section: Realizability Preserving Discretization Of the Moment Equationsmentioning

confidence: 99%

A numerical comparison of the method of moments for the population balance equation

Müller

Klar

Schneider

2019

Mathematics and Computers in Simulation

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We investigate the application of the method of moments approach for the one-dimensional population balance equation. We consider different types of moment closures, namely polynomial (P N ) closures, maximum entropy (M N ) closures and the quadrature method of moments QM OM N . Realizability issues and implementation details are discussed. The numerical examples range from spatially homogeneous cases to a population balance equation coupled with fluid dynamic equations for a lid-driven cavity test case. A detailed numerical discussion of accuracy, order of the moment method and computational time is given.Keywords: population balance, maximum entropy, quadrature method of moments 2010 MSC: 35L40, 45K05, 35R09 IntroductionPopulation balance equations are widely used in engineering applications, including aerosol physics, high shear granulation, pharmaceutical industries, polymerization and emulsion processes, evaporation and condensation processes in bubble column reactors, bioreactors, turbulent flame reactors and many others, see [13,19,24,24,[43][44][45] and references therein. These polydisperse processes are characterized by two phases: one of them is the continuous and the second is a dispersed phase consisting of particles. The particles can take different forms like crystals, drops or bubbles with several possible properties such as volume, chemical composition, porosity and enthalpy. In this work only the volume is considered. The dynamic evolution of the particle number distribution which is described by the population balance equation (PBE) of the dispersed phase depends not only on the particle-particle interactions, but also on the continuous phase, due to interaction of these particles with the continuous flow field in which they are dispersed. These interactions usually result in the common mechanisms aggregation, breakage, condensation, growth and nucleation. In our work we concentrate on binary aggregation, namely the Smoluchowski aggregation [42] and multiple breakages, since binary breakage is not sufficient for some of these applications. Binary aggregation is the process of merging two particles to a larger particle, whereas in a breakage process, a particle breaks into several smaller fragments. Often included in a typical PBE are spatial transport terms, i.e. advection and diffusion terms. We use the simplest case and assume that the mean particle velocity is the same as those of the fluid. The resulting population balance equations range from integro-differential to partial integro-differential equations of hyperbolic or parabolic type in phase space, in addition to the differential terms for advection and diffusion in the physical space.

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“…Here γ ∈ (0, ∞) is a parameter and · is the usual Euclidean norm on R n . Unlike the original primal problem (15), the relaxed problem (35) is feasible for any v ∈ R n (not just v ∈ R), so we expect that it will have a solution for most, indeed perhaps all, v ∈ R n . Whenever a solution to (35) exists, it has the same form as that of the original primal problem:…”

Section: Realizability and Relaxation Of The Entropy Minimization Promentioning

confidence: 99%

“…In this section, we propose a new set of closures, based on the regularization (35). We replace (21) by the system of regularized entropy-based moment equations…”

Section: Regularized Entropy-based Closuresmentioning

confidence: 99%

A Regularized Entropy-Based Moment Method for Kinetic Equations

Alldredge¹,

Frank²,

Hauck³

2019

SIAM J. Appl. Math.

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The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.

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A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry

Cited by 24 publications

References 38 publications

A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations

A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations

A numerical comparison of the method of moments for the population balance equation

A Regularized Entropy-Based Moment Method for Kinetic Equations

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