Adaptive Sparse-grid Discontinuous Galerkin Approximations to the Bhatnagar--Gross--Krook Model
Abstract
This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar--Gross--Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.
Related
- The sharp CFL condition of the piecewise constant sparse grid discontinuous Galerkin method for high-dimensional transport equations
- Adaptive Multilevel Discontinuous Galerkin Methods on GPUs
- Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme
- Adaptive local representations for Helmholtz Trefftz discontinuous Galerkin methods
- Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
BibTeX
@misc{schnake2026adaptive,
title = {{Adaptive Sparse-grid Discontinuous Galerkin Approximations to the Bhatnagar--Gross--Krook Model}},
author = {Stefan Schnake and Miroslav Stoyanov and Eirik Endeve and Cory Hauck},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20395},
eprint = {2609.20395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20395},
}
RIS
TY - UNPB
TI - Adaptive Sparse-grid Discontinuous Galerkin Approximations to the Bhatnagar--Gross--Krook Model
AU - Stefan Schnake
AU - Miroslav Stoyanov
AU - Eirik Endeve
AU - Cory Hauck
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.20395
UR - https://arxiv.org/abs/2609.20395
AB - This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar--Gross--Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.
ER -