Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Schnake, Stefan; Kendrick, Coleman; Endeve, Eirik; Stoyanov, Miroslav; Hahn, Steven; Hauck, Cory D.; Green, David L.; Snyder, Phil; Canik, John

doi:10.1016/j.jcp.2024.113053

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Journal Article · Tue May 07 00:00:00 EDT 2024 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2024.113053· OSTI ID:2394732

^[1]; ^[2]; ^[3]; ^[1]; ^[1]; ^[3]; ^[4]; Snyder, Phil ^[1]; ^[5]

Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Univ. of Tennessee, Knoxville, TN (United States)
Commonwealth Scientific and Industrial Research Organisation (CSIRO), Mayfield West, NSW (Australia)
Type One Energy Group, Oak Ridge, TN (United States)

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. Here we use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: AC05-00OR22725; AC52-07NA27344

OSTI ID:: 2394732

Alternate ID(s):: OSTI ID: 2476897

Report Number(s):: LLNL--JRNL-860778

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 510; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (65)

A Critical Comparison of Eulerian-Grid-Based Vlasov Solvers Arber, T. D.; Vann, R. G. L. Journal of Computational Physics, Vol. 180, Issue 1 https://doi.org/10.1006/jcph.2002.7098	journal	July 2002
Sparse-grid finite-volume multigrid for 3D-problems Hemker, P. W. Advances in Computational Mathematics, Vol. 4, Issue 1 https://doi.org/10.1007/BF02123474	journal	December 1995
A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations Cheng, Yingda; Gamba, Irene M.; Majorana, Armando SeMA Journal, Vol. 54, Issue 1 https://doi.org/10.1007/BF03322587	journal	April 2011
Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs Nobile, F.; Tamellini, L.; Tempone, R. Numerische Mathematik, Vol. 134, Issue 2 https://doi.org/10.1007/s00211-015-0773-y	journal	October 2015
High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs Chkifa, Abdellah; Cohen, Albert; Schwab, Christoph Foundations of Computational Mathematics, Vol. 14, Issue 4 https://doi.org/10.1007/s10208-013-9154-z	journal	May 2013
Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws Friedrich, Lucas; Schnücke, Gero; Winters, Andrew R. Journal of Scientific Computing, Vol. 80, Issue 1 https://doi.org/10.1007/s10915-019-00933-2	journal	March 2019
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws Sod, Gary A. Journal of Computational Physics, Vol. 27, Issue 1 https://doi.org/10.1016/0021-9991(78)90023-2	journal	April 1978
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II larsen, Edward W.; Morel, J. E. Journal of Computational Physics, Vol. 83, Issue 1 https://doi.org/10.1016/0021-9991(89)90229-5	journal	July 1989
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations Ascher, Uri M.; Ruuth, Steven J.; Spiteri, Raymond J. Applied Numerical Mathematics, Vol. 25, Issue 2-3 https://doi.org/10.1016/S0168-9274(97)00056-1	journal	November 1997
Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation laws Yan, Ge; Kaur, Sharanjeet; Banks, Jeffrey W. Journal of Computational and Applied Mathematics, Vol. 422 https://doi.org/10.1016/j.cam.2022.114885	journal	April 2023
A dynamically adaptive sparse grids method for quasi-optimal interpolation of multidimensional functions Stoyanov, Miroslav K.; Webster, Clayton G. Computers & Mathematics with Applications, Vol. 71, Issue 11 https://doi.org/10.1016/j.camwa.2015.12.045	journal	June 2016
An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation Abdelmalik, M. R. A.; van Brummelen, E. H. Computers & Mathematics with Applications, Vol. 72, Issue 8 https://doi.org/10.1016/j.camwa.2016.05.021	journal	October 2016
On discontinuous Galerkin approximations of Boltzmann moment systems with Levermore closure Barth, Timothy Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 25-28 https://doi.org/10.1016/j.cma.2005.06.016	journal	May 2006
Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting Pazner, Will Computer Methods in Applied Mechanics and Engineering, Vol. 382 https://doi.org/10.1016/j.cma.2021.113876	journal	August 2021
hPIC: A scalable electrostatic Particle-in-Cell for Plasma–Material Interactions Khaziev, Rinat; Curreli, Davide Computer Physics Communications, Vol. 229 https://doi.org/10.1016/j.cpc.2018.03.028	journal	August 2018
Spatially adaptive sparse grids for high-dimensional data-driven problems Pflüger, Dirk; Peherstorfer, Benjamin; Bungartz, Hans-Joachim Journal of Complexity, Vol. 26, Issue 5 https://doi.org/10.1016/j.jco.2010.04.001	journal	October 2010
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics Bennoune, Mounir; Lemou, Mohammed; Mieussens, Luc Journal of Computational Physics, Vol. 227, Issue 8 https://doi.org/10.1016/j.jcp.2007.11.032	journal	April 2008
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Ma, Xiang; Zabaras, Nicholas Journal of Computational Physics, Vol. 228, Issue 8 https://doi.org/10.1016/j.jcp.2009.01.006	journal	May 2009
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources Filbet, Francis; Jin, Shi Journal of Computational Physics, Vol. 229, Issue 20 https://doi.org/10.1016/j.jcp.2010.06.017	journal	October 2010
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids Jakeman, John D.; Archibald, Richard; Xiu, Dongbin Journal of Computational Physics, Vol. 230, Issue 10 https://doi.org/10.1016/j.jcp.2011.02.022	journal	May 2011
An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm Chen, G.; Chacón, L.; Barnes, D. C. Journal of Computational Physics, Vol. 230, Issue 18 https://doi.org/10.1016/j.jcp.2011.05.031	journal	August 2011
Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system Qiu, Jing-Mei; Shu, Chi-Wang Journal of Computational Physics, Vol. 230, Issue 23 https://doi.org/10.1016/j.jcp.2011.07.018	journal	September 2011
A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer Olbrant, Edgar; Hauck, Cory D.; Frank, Martin Journal of Computational Physics, Vol. 231, Issue 17 https://doi.org/10.1016/j.jcp.2012.03.002	journal	July 2012
Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions Jakeman, John D.; Narayan, Akil; Xiu, Dongbin Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.02.035	journal	June 2013
Energy-conserving discontinuous Galerkin methods for the Vlasov–Ampère system Cheng, Yingda; Christlieb, Andrew J.; Zhong, Xinghui Journal of Computational Physics, Vol. 256 https://doi.org/10.1016/j.jcp.2013.09.013	journal	January 2014
High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation Xiong, Tao; Jang, Juhi; Li, Fengyan Journal of Computational Physics, Vol. 284 https://doi.org/10.1016/j.jcp.2014.12.021	journal	March 2015
Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates Endeve, Eirik; Hauck, Cory D.; Xing, Yulong Journal of Computational Physics, Vol. 287 https://doi.org/10.1016/j.jcp.2015.02.005	journal	April 2015
Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations Wang, Zixuan; Tang, Qi; Guo, Wei Journal of Computational Physics, Vol. 314 https://doi.org/10.1016/j.jcp.2016.03.005	journal	June 2016
An asymptotic preserving method for strongly anisotropic diffusion equations based on field line integration Tang, Min; Wang, Yihong Journal of Computational Physics, Vol. 330 https://doi.org/10.1016/j.jcp.2016.10.062	journal	February 2017
Realizability-preserving DG-IMEX method for the two-moment model of fermion transport Chu, Ran; Endeve, Eirik; Hauck, Cory D. Journal of Computational Physics, Vol. 389 https://doi.org/10.1016/j.jcp.2019.03.037	journal	July 2019
Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems Fu, Guosheng; Shu, Chi-Wang Journal of Computational Physics, Vol. 394 https://doi.org/10.1016/j.jcp.2019.05.050	journal	October 2019
A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps Guo, Wei; Qiu, Jing-Mei Journal of Computational Physics, Vol. 458 https://doi.org/10.1016/j.jcp.2022.111089	journal	June 2022
A generalized Eulerian-Lagrangian discontinuous Galerkin method for transport problems Hong, Xue; Qiu, Jing-Mei Journal of Computational Physics, Vol. 464 https://doi.org/10.1016/j.jcp.2022.111160	journal	September 2022
Conservative DG method for the micro-macro decomposition of the Vlasov–Poisson–Lenard–Bernstein model Endeve, Eirik; Hauck, Cory D. Journal of Computational Physics, Vol. 462 https://doi.org/10.1016/j.jcp.2022.111227	journal	August 2022
Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators Hakim, Ammar; Francisquez, Manaure; Juno, James Journal of Plasma Physics, Vol. 86, Issue 4 https://doi.org/10.1017/S0022377820000586	journal	July 2020
Sparse grids Bungartz, Hans-Joachim; Griebel, Michael Acta Numerica, Vol. 13 https://doi.org/10.1017/S0962492904000182	journal	May 2004
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Sparse finite element approximation of high-dimensional transport-dominated diffusion problems Schwab, Christoph; Süli, Endre; Todor, Radu Alexandru ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 42, Issue 5 https://doi.org/10.1051/m2an:2008027	journal	July 2008
Kinetic-Theoretic Description of the Formation of a Shock Wave Chu, C. K. The Physics of Fluids, Vol. 8, Issue 1 https://doi.org/10.1063/1.1761077	journal	January 1965
Generalized weighting scheme for δ f particle‐simulation method Hu, Genze; Krommes, John A. Physics of Plasmas, Vol. 1, Issue 4 https://doi.org/10.1063/1.870745	journal	April 1994
An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation Alekseenko, A.; Gimelshein, N.; Gimelshein, S. International Journal of Computational Fluid Dynamics, Vol. 26, Issue 3 https://doi.org/10.1080/10618562.2012.668889	journal	March 2012
Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator Francisquez, M.; Bernard, T. N.; Mandell, N. R. Nuclear Fusion, Vol. 60, Issue 9 https://doi.org/10.1088/1741-4326/aba0c9	journal	August 2020
Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data Mishra, S.; Schwab, Ch. Mathematics of Computation, Vol. 81, Issue 280 https://doi.org/10.1090/S0025-5718-2012-02574-9	journal	April 2012
Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering Sheng, Qiwei; Hauck, Cory D. Mathematics of Computation, Vol. 90, Issue 332 https://doi.org/10.1090/mcom/3670	journal	July 2021
Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry Zhang, Weijie; Xing, Yulong; Endeve, Eirik Monthly Notices of the Royal Astronomical Society, Vol. 514, Issue 1 https://doi.org/10.1093/mnras/stac1257	journal	May 2022
Plasma Oscillations with Diffusion in Velocity Space Lenard, A.; Bernstein, Ira B. Physical Review, Vol. 112, Issue 5 https://doi.org/10.1103/PhysRev.112.1456	journal	December 1958
On adaptive control processes Bellman, R.; Kalaba, R. IRE Transactions on Automatic Control, Vol. 4, Issue 2 https://doi.org/10.1109/TAC.1959.1104847	journal	November 1959
A Class of Bases in $L^2$ for the Sparse Representation of Integral Operators Alpert, Bradley K. SIAM Journal on Mathematical Analysis, Vol. 24, Issue 1 https://doi.org/10.1137/0524016	journal	January 1993
Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit Guermond, Jean-Luc; Kanschat, Guido SIAM Journal on Numerical Analysis, Vol. 48, Issue 1 https://doi.org/10.1137/090746938	journal	January 2010
Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross Shen, Jie; Wang, Li-Lian SIAM Journal on Numerical Analysis, Vol. 48, Issue 3 https://doi.org/10.1137/090765547	journal	January 2010
Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems Shen, Jie; Yu, Haijun SIAM Journal on Scientific Computing, Vol. 32, Issue 6 https://doi.org/10.1137/100787842	journal	January 2010
Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation Narayan, Akil; Jakeman, John D. SIAM Journal on Scientific Computing, Vol. 36, Issue 6 https://doi.org/10.1137/140966368	journal	January 2014
A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations Guo, Wei; Cheng, Yingda SIAM Journal on Scientific Computing, Vol. 38, Issue 6 https://doi.org/10.1137/16M1060017	journal	January 2016
An Adaptive Multiresolution Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multidimensions Guo, Wei; Cheng, Yingda SIAM Journal on Scientific Computing, Vol. 39, Issue 6 https://doi.org/10.1137/16M1083190	journal	January 2017
A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework Garrett, C. Kristopher; Hauck, Cory D. SIAM Journal on Scientific Computing, Vol. 40, Issue 2 https://doi.org/10.1137/17M1134184	journal	January 2018
A Low-Rank Projector-Splitting Integrator for the Vlasov--Poisson Equation Einkemmer, Lukas; Lubich, Christian SIAM Journal on Scientific Computing, Vol. 40, Issue 5 https://doi.org/10.1137/18M116383X	journal	January 2018
Efficient Entropy Stable Gauss Collocation Methods Chan, Jesse; Del Rey Fernández, David C.; Carpenter, Mark H. SIAM Journal on Scientific Computing, Vol. 41, Issue 5 https://doi.org/10.1137/18M1209234	journal	January 2019
An Adaptive Multiresolution Discontinuous Galerkin Method with Artificial Viscosity for Scalar Hyperbolic Conservation Laws in Multidimensions Huang, Juntao; Cheng, Yingda SIAM Journal on Scientific Computing, Vol. 42, Issue 5 https://doi.org/10.1137/19M126565X	journal	January 2020
A Method for Dimensionally Adaptive Sparse Trigonometric Interpolation of Periodic Functions Morrow, Zachary; Stoyanov, Miroslav SIAM Journal on Scientific Computing, Vol. 42, Issue 4 https://doi.org/10.1137/19M1283483	journal	January 2020
An Efficient Dynamical Low-Rank Algorithm for the Boltzmann-BGK Equation Close to the Compressible Viscous Flow Regime Einkemmer, Lukas; Hu, Jingwei; Ying, Lexing SIAM Journal on Scientific Computing, Vol. 43, Issue 5 https://doi.org/10.1137/21M1392772	journal	January 2021
An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model DeCaria, Victor P.; Hauck, Cory D.; Schnake, Stefan R. SIAM Journal on Numerical Analysis, Vol. 62, Issue 3 https://doi.org/10.1137/22M1485784	journal	May 2024
The Solution of Multidimensional Real Helmholtz Equations on Sparse Grids Balder, Robert; Zenger, Christoph SIAM Journal on Scientific Computing, Vol. 17, Issue 3 https://doi.org/10.1137/S1064827593247035	journal	May 1996
Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems Adams, Marvin L. Nuclear Science and Engineering, Vol. 137, Issue 3 https://doi.org/10.13182/NSE00-41	journal	March 2001
Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles Crestetto, Anaïs; Crouseilles, Nicolas; Lemou, Mohammed Kinetic & Related Models, Vol. 5, Issue 4 https://doi.org/10.3934/krm.2012.5.787	journal	January 2012
Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes Tianheng Chen, Tianheng Chen; Chi-Wang Shu, Chi-Wang Shu CSIAM Transactions on Applied Mathematics, Vol. 1, Issue 1 https://doi.org/10.4208/csiam-am.2020-0003	journal	January 2020

Similar Records

Adaptive Discontinuous Galerkin Methods in Multiwavelets Bases

Journal Article · Fri Dec 31 23:00:00 EST 2010 · Applied Numerical Mathematics · OSTI ID:1008820

Conservative DG method for the micro-macro decomposition of the Vlasov–Poisson–Lenard–Bernstein model

Journal Article · Thu Apr 14 20:00:00 EDT 2022 · Journal of Computational Physics · OSTI ID:2538513

A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system

Journal Article · Fri May 20 00:00:00 EDT 2011 · AIP Conference Proceedings · OSTI ID:21511592

Related Subjects

97 MATHEMATICS AND COMPUTING
Lenard-Bernstein
Vlasov-Poisson
discontinuous Galerkin
implicit-explicit
kinetic equation
sparse grids

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Citation Formats

References (65)

Similar Records

Related Subjects