A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations

Allmann-Rahn, F.; Grauer, Rainer; Kormann, Katharina

doi:10.1016/j.jcp.2022.111562

Cited by 9 publications

(7 citation statements)

References 36 publications

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“…onto the space spanned by the V f j ). That is, we have ∂ t K f j = ⟨V f j , RHS⟩ zv , where RHS is the right-hand side of equation (1). Thus, we have…”

Section: Equation For Kmentioning

confidence: 98%

“…The equation for the vector potential is similar. However, in the evolution equation (1) we actually need the time derivative, i.e. ∂ t A, and not the vector potential directly.…”

Section: Model Equationmentioning

confidence: 99%

“…In all of these cases ξ 1 contains the spatial variables and ξ 2 contains all the velocity variables. This reduces a six (or five) dimensional problem to a three-dimensional problem (although tensor approximations that further reduce the dimensionality of the problem have also been proposed [36,19,1,28]). Besides numerical evidence, there are also physical reasons why such an approximation is effective.…”

Section: Dynamical Low-rank Algorithmmentioning

confidence: 99%

“…While such methods can be applied in a rather generic way to ordinary or partial differential equation, an efficient algorithm is only obtained if a suitable decomposition of variables is chosen that allows us to run the simulation with a small to moderate rank. For kinetic problems primarily a decomposition between spatial and velocity variables has been performed (see [16,22,23,49,48,12,8,38]; some work on tensor decomposition also exists [36,19,1,28]). It turns out that for kinetic problems this has a number of advantages.…”

Section: Introductionmentioning

confidence: 99%

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A robust and conservative dynamical low-rank algorithm

Einkemmer

Ostermann

Scalone

2023

Journal of Computational Physics

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“…onto the space spanned by the V f j ). That is, we have ∂ t K f j = ⟨V f j , RHS⟩ zv , where RHS is the right-hand side of equation (1). Thus, we have…”

Section: Equation For Kmentioning

confidence: 98%

“…The equation for the vector potential is similar. However, in the evolution equation (1) we actually need the time derivative, i.e. ∂ t A, and not the vector potential directly.…”

Section: Model Equationmentioning

confidence: 99%

Section: Dynamical Low-rank Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A robust and conservative dynamical low-rank algorithm

Einkemmer

Ostermann

Scalone

2023

Journal of Computational Physics

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“…In [33,34], the Tensor Train representation [35] was used to obtain a reduced-order representation of the velocity distribution function, with the approach differing from most other works in that the reduced-order representation of the velocity distribution function was stored independently for each cell on the physical space grid, which allows for the use of unstructured grids, whereas in previous studies [36,37,27], the decomposition was applied to the multi-dimensional Cartesian product of the (structured) physical and velocity domains. A similar approach was utilized later on in [38] for the Vlasov-Maxwell equations, with low-rank decomposition applied only in velocity space and not the full 6-dimensional physical and phase space.…”

mentioning

confidence: 99%

Modeling of Ionized Gas Flows with a Velocity-space Hybrid Boltzmann Solver

Oblapenko

Goldstein

Varghese

et al. 2021

AIAA Scitech 2021 Forum

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In the present work, the Tensor-Train decomposition algorithm is applied to reduce the memory footprint of a stochastic discrete velocity solver for rarefied gas dynamics simulation. An energyconserving modification to the algorithm is proposed, along with an interleaved collision/convection routine which allows for easy application of higher-order convection schemes. The performance of the developed algorithm is analyzed for several 0-and 1-dimensional model problems in terms of solution error and reduction in memory use requirements.Keywords Boltzmann equation • discrete velocity method • tensor decomposition • tensor train • rarefied gas • low-rank approximation IntroductionThe motion of gas or fluid can be successfully described by a set of partial different equations under a wide range of conditions of interest. However, in rarefied regimes, when the ratio of the molecular mean free path to the characteristic length scale becomes large, the continuum approaches no longer apply, and instead, the full Boltzmann equation describing the evolution of the velocity distribution function due to advection, external forces, and collisions, has to be solved. It is a complicated integro-differential equation, with a 6-dimensional integral (the collision operator) appearing on the right-hand side. Numerous approaches have been developed over the years to tackle the equation. These include stochastic approaches, such as the Direct Simulation Monte Carlo (DSMC) method [1], and deterministic approaches, such as discrete velocity methods [2] and spectral methods [3,4].The family of discrete velocity methods (DVM), which is the main focus of the present work, operates by discretizing the velocity distribution function on a grid in velocity space, and obtains a system of partial differential equations for the values of the distribution function at each grid node. A significant advantage offered by discrete velocity methods is the noticeable reduction in noise compared to particle-based methods, which is needed to better understand the small time-scale dynamics of complex rarefied flows, especially in unsteady scenarios [5,6,7].Within the discrete velocity framework, the complex collision operator is often replaced with a simple relaxation term [8,9]. Such relaxation terms include the Bhatnagar-Gross-Krook (BGK) model [10] and its extensions, the ellipsoidal-statistical BGK model (ES-BGK) [11] and the Shakhov model (S-model) [12]. However, the correct incorporation of complex collisional physical phenomena, such as chemical reactions, transitions of internal energy, and complicated scattering laws within the framework of these model equations is still an open question and a topic of active research [13,14,15]. Despite these drawbacks of the linear models, they offer advantages in terms of being scalable to dense regimes [16,17].Another subfamily of the discrete velocity methods does not resort to substituting the collision operator with a simplified model relaxation term, and instead models the full collision process, utilizing...

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Introduction to the numerical flow iteration for the Vlasov–Poisson equation

Wilhelm,

Kirchhart,

Torrilhon

2023

Proc Appl Math and Mech

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We present a novel approach to solve the Vlasov–Poisson equation: the numerical flow iteration. It evaluates the numerical solution by storing the low‐dimensional electric potentials and uses these to iteratively reconstruct the characteristics. This reduces the total memory footprint by several orders of magnitude as complexity gets shifted from memory access to computation on‐the‐fly. Furthermore, this ansatz yields strong conservation properties due to the exploitation of the solution structure.

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A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations

Cited by 9 publications

References 36 publications

A robust and conservative dynamical low-rank algorithm

A robust and conservative dynamical low-rank algorithm

Modeling of Ionized Gas Flows with a Velocity-space Hybrid Boltzmann Solver

Introduction to the numerical flow iteration for the Vlasov–Poisson equation

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