Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions

Ritter, Juliane; Klar, Axel; Schneider, Florian

doi:10.1016/j.cam.2016.04.019

Cited by 12 publications

(10 citation statements)

References 32 publications

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“…They are based on a partition of the velocity space, or, equivalently, based on basis functions with local support. While the partial-moment models have been extensively studied for special cases (like half-or quarter-moments in one or two dimensions) [58,60,142,159,160], we are unaware of any general investigation, especially in the fully three-dimensional setup.…”

Section: Angular Bases In Slab Geometrymentioning

confidence: 99%

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

Schneider

Leibner

2020

Journal of Computational Physics

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Moment models are a class of specialized approximate models for kinetic transport equations. These models transform the kinetic equation to a system of equations for weighted velocity averages of the solution, called moments, thereby removing the velocity dependency. The properties of the resulting models depend on the chosen weight functions for the moments and on the approach used to close the equations. Closing the system by specifying a linear ansatz function results in linear models that are comparatively easy to solve but may show non-physical behaviour. Minimum-entropy moment closures, on the other hand, conserve many of the fundamental physical properties but require the solution of a non-linear optimization problem for every cell in the space-time grid. In addition, these models are only well-defined if the moments can be kept within a particular subset of the real coordinate space, the so-called realizable set, which is particularly challenging for higher-order numerical solvers.Schließlich beschäftigen wir uns mit der Möglichkeit einer zusätzlichen Modellreduktion für parameterabhängige Momentenmodelle. Hierzu nutzen wir die Reduzierte-Basis-Methode und berechnen mit der Hauptkomponentenanalyse ("Proper Orthogonal Decomposition", POD) ein reduziertes Modell. Wir stellen die hierarchische approximative POD (HAPOD) vor, ein universelles und einfach zu implementierendes Verfahren um eine approximative POD zu berechnen. Die HAPOD nutzt eine fast beliebig an die verfügbaren Ressourcen anpassbare Baumstruktur, so dass jeder Rechenknoten nur die POD einer relativ kleinen Teilmenge der Eingangsvektoren berechnen muss. Wir führen eine ausführliche theoretische Analyse der HAPOD durch und zeigen die Anwendbarkeit auf lineare Momentenmodelle sowie eine gegenüber der POD deutlich verbesserte Effizienz.

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Section: Angular Bases In Slab Geometrymentioning

confidence: 99%

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

Schneider

Leibner

2020

Journal of Computational Physics

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“…In recent years many modifications to this closure have been suggested, including the positive P N (PP N ), filtered P N (FP N ) and diffusive-corrected P N (D N ) [38], curing some of the disadvantages of the original P N method while increasing the complexity of the system at the price of higher computational costs. We also want to note that the choices of other closures and angular bases are possible, e.g., minimum entropy [3,31,39,[39][40][41][42][43][44][45][46][47][48][49], partial and mixed moments [50][51][52][53][54][55] or Kershaw closures [56][57][58][59].…”

Section: Moment Approximationsmentioning

confidence: 99%

The second-order formulation of the $P_N$ equations with Marshak boundary conditions

Andres,

Schneider

2019

Preprint

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We consider a reformulation of the classical P N method with Marshak boundary conditions for the approximation of the monoenergetic stationary linear transport equation as a system of second-order PDEs. Our derivation allows the automatic generation of a model hierarchy which can then be handed to standard PDE tools. This method allows for heterogeneous coefficients, irregular grids, anisotropic boundary sources and anisotropic scattering. The wide applicability is demonstrated in several numerical test cases. We make our implementation available online, which allows for fast prototyping.

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“…Checking realizability is much easier when using piecewise linear bases instead of the standard polynomial basis on the whole velocity space [19,20,49,59,[62][63][64]. In addition, the computational cost is significantly lower for these models.…”

Section: Introductionmentioning

confidence: 99%

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

Leibner

Ohlberger

2021

ESAIM: M2AN

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In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

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Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions

Cited by 12 publications

References 32 publications

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: Model derivation and realizability theory

The second-order formulation of the $P_N$ equations with Marshak boundary conditions

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

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