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Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition (2311.09425v2)

Published 15 Nov 2023 in math.NA, cs.NA, and physics.comp-ph

Abstract: Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

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References (41)
  1. Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. Journal of Computational Physics, 227(8):3781–3803, April 2008. ISSN 00219991. doi:10.1016/j.jcp.2007.11.032.
  2. Plasma Physics via Computer Simulation. McGraw-Hill, New York, 1985. ISBN 978-0-07-005371-7.
  3. An unconventional robust integrator for dynamical low-rank approximation. BIT Numerical Mathematics, 62(1):23–44, March 2022. ISSN 0006-3835, 1572-9125. doi:10.1007/s10543-021-00873-0.
  4. A robust second-order low-rank BUG integrator based on the midpoint rule. February 2024. doi:10.48550/arXiv.2402.08607.
  5. Efficient dynamical low-rank approximation for the Vlasov-Ampère-Fokker-Planck system. Journal of Computational Physics, page 111590, September 2022. ISSN 00219991. doi:10.1016/j.jcp.2022.111590.
  6. G.L. Delzanno. Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form. Journal of Computational Physics, 301:338–356, November 2015. ISSN 00219991. doi:10.1016/j.jcp.2015.07.028.
  7. J. P. Dougherty. Model Fokker-Planck Equation for a Plasma and Its Solution. Physics of Fluids, 7(11):1788, 1964. ISSN 00319171. doi:10.1063/1.2746779.
  8. Lukas Einkemmer. Accelerating the simulation of kinetic shear Alfvén waves with a dynamical low-rank approximation. 2023. doi:10.48550/ARXIV.2306.17526.
  9. A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation. Journal of Computational Physics, 443:110495, October 2021. ISSN 00219991. doi:10.1016/j.jcp.2021.110495.
  10. A Low-Rank Projector-Splitting Integrator for the Vlasov–Poisson Equation. SIAM Journal on Scientific Computing, 40(5):B1330–B1360, January 2018. ISSN 1064-8275, 1095-7197. doi:10.1137/18M116383X.
  11. A Quasi-Conservative Dynamical Low-Rank Algorithm for the Vlasov Equation. SIAM Journal on Scientific Computing, 41(5):B1061–B1081, January 2019. ISSN 1064-8275, 1095-7197. doi:10.1137/18M1218686.
  12. An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation. Journal of Computational Physics, 439:110353, August 2021a. ISSN 00219991. doi:10.1016/j.jcp.2021.110353.
  13. An Efficient Dynamical Low-Rank Algorithm for the Boltzmann-BGK Equation Close to the Compressible Viscous Flow Regime. SIAM Journal on Scientific Computing, 43(5):B1057–B1080, January 2021b. ISSN 1064-8275, 1095-7197. doi:10.1137/21M1392772.
  14. Conservation properties of the augmented basis update & Galerkin integrator for kinetic problems. November 2023a. doi:10.48550/arXiv.2311.06399.
  15. A robust and conservative dynamical low-rank algorithm. Journal of Computational Physics, 484:112060, July 2023b. ISSN 00219991. doi:10.1016/j.jcp.2023.112060.
  16. Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov–Poisson System. Communications on Applied Mathematics and Computation, 4(1):34–59, March 2022. ISSN 2096-6385, 2661-8893. doi:10.1007/s42967-020-00089-z.
  17. Walter Gautschi. Orthogonal Polynomials: Computation and Approximation. Oxford University Press, April 2004. ISBN 978-0-19-850672-0 978-0-19-191657-1. doi:10.1093/oso/9780198506720.001.0001.
  18. Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. WORLD SCIENTIFIC, January 2011. ISBN 978-981-4289-26-9 978-981-4289-27-6. doi:10.1142/7498.
  19. Harold Grad. On the kinetic theory of rarefied gases. Communications on Pure and Applied Mathematics, 2(4):331–407, December 1949. ISSN 00103640, 10970312. doi:10.1002/cpa.3160020403.
  20. A conservative low rank tensor method for the Vlasov dynamics. January 2022a. doi:10.48550/arXiv.2201.10397.
  21. A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov dynamics. July 2022b. doi:10.48550/arXiv.2207.00518.
  22. A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps. Journal of Computational Physics, 458:111089, June 2022c. ISSN 00219991. doi:10.1016/j.jcp.2022.111089.
  23. A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics. Communications on Applied Mathematics and Computation, July 2023. ISSN 2096-6385, 2661-8893. doi:10.1007/s42967-023-00277-7.
  24. Alias-Free, Matrix-Free, and Quadrature-Free Discontinuous Galerkin Algorithms for (Plasma) Kinetic Equations. In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–15, Atlanta, GA, USA, November 2020. IEEE. ISBN 978-1-72819-998-6. doi:10.1109/SC41405.2020.00077.
  25. Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators. Journal of Plasma Physics, 86(4):905860403, August 2020. ISSN 0022-3778, 1469-7807. doi:10.1017/S0022377820000586.
  26. Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations. Frontiers in Physics, 6:105, September 2018. ISSN 2296-424X. doi:10.3389/fphy.2018.00105.
  27. Computing nearly singular solutions using pseudo-spectral methods. Journal of Computational Physics, 226(1):379–397, September 2007. ISSN 00219991. doi:10.1016/j.jcp.2007.04.014.
  28. An Adaptive Dynamical Low Rank Method for the Nonlinear Boltzmann Equation. Journal of Scientific Computing, 92(2):75, August 2022. ISSN 0885-7474, 1573-7691. doi:10.1007/s10915-022-01934-4.
  29. Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values. SIAM Journal on Numerical Analysis, 54(2):1020–1038, January 2016. ISSN 0036-1429, 1095-7170. doi:10.1137/15M1026791.
  30. Dynamical Low-Rank Approximation. SIAM Journal on Matrix Analysis and Applications, 29(2):434–454, January 2007. ISSN 0895-4798, 1095-7162. doi:10.1137/050639703.
  31. Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation. February 2023.
  32. The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov–Maxwell equations. Computer Physics Communications, 264:107866, July 2021. ISSN 00104655. doi:10.1016/j.cpc.2021.107866.
  33. A projector-splitting integrator for dynamical low-rank approximation. BIT Numerical Mathematics, 54(1):171–188, March 2014. ISSN 0006-3835, 1572-9125. doi:10.1007/s10543-013-0454-0.
  34. Randomized Numerical Linear Algebra: Foundations & Algorithms. arXiv:2002.01387 [cs, math], August 2020.
  35. A three-dimensional finite-volume solver for the Maxwell equations with divergence cleaning on unstructured meshes. Computer Physics Communications, 130(1-2):83–117, July 2000. ISSN 00104655. doi:10.1016/S0010-4655(00)00045-X.
  36. NIST digital library of mathematical functions, 2020.
  37. A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations. Journal of Computational Physics, 447:110672, December 2021. ISSN 00219991. doi:10.1016/j.jcp.2021.110672.
  38. A low-rank method for two-dimensional time-dependent radiation transport calculations. Journal of Computational Physics, 421:109735, November 2020. ISSN 00219991. doi:10.1016/j.jcp.2020.109735.
  39. Chi-Wang Shu. Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, volume 1697, pages 325–432. Springer Berlin Heidelberg, Berlin, Heidelberg, 1998. ISBN 978-3-540-64977-9 978-3-540-49804-9. doi:10.1007/BFb0096355.
  40. Chi-Wang Shu. High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems. SIAM Review, 51(1):82–126, February 2009. ISSN 0036-1445, 1095-7200. doi:10.1137/070679065.
  41. SpectralPlasmaSolver: A Spectral Code for Multiscale Simulations of Collisionless, Magnetized Plasmas. Journal of Physics: Conference Series, 719:012022, May 2016. ISSN 1742-6588, 1742-6596. doi:10.1088/1742-6596/719/1/012022.
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