A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations
A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations
View/Date of Publication
09.2022Publisher
Institut für Numerische SimulationPublisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11581
Abstract
The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.
The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov–Poisson solver show a stabilizing effect of the new bases on the simulations.
The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov–Poisson solver show a stabilizing effect of the new bases on the simulations.
Subjects
Vlasov–Poisson equations, Sparse grid combination technique, Multi-scale functions, Numerical instabilities, Kinetic simulations
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.48550/arXiv.2209.14064Part of Series
INS Preprints ; 2209Pollinger, Theresa; Rentrop, Johannes; Pflüger, Dirk; Kormann, Katharina: A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations. In: INS Preprints, 2209.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11581
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11581
@unpublished{handle:20.500.11811/11581,
author = {{Theresa Pollinger} and {Johannes Rentrop} and {Dirk Pflüger} and {Katharina Kormann}},
title = {A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations},
publisher = {Institut für Numerische Simulation},
year = 2022,
month = sep,
INS Preprints},
volume = 2209,
note = {The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.
The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov–Poisson solver show a stabilizing effect of the new bases on the simulations.},
url = {https://hdl.handle.net/20.500.11811/11581}
}
author = {{Theresa Pollinger} and {Johannes Rentrop} and {Dirk Pflüger} and {Katharina Kormann}},
title = {A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations},
publisher = {Institut für Numerische Simulation},
year = 2022,
month = sep,
INS Preprints},
volume = 2209,
note = {The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.
The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov–Poisson solver show a stabilizing effect of the new bases on the simulations.},
url = {https://hdl.handle.net/20.500.11811/11581}
}
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