Methods and Algorithms for Scientific Computing

A Low-Rank Algorithm for Weakly Compressible Flow


In this paper, we propose a numerical method for solving weakly compressible fluid flow based on a dynamical low-rank projector splitting. The low-rank splitting scheme is applied to the Boltzmann equation with BGK collision term, which results in a set of constant-coefficient advection equations. This procedure is numerically efficient as a small rank is sufficient to obtain the relevant dynamics (described by the Navier--Stokes equations). The resulting method can be combined with a range of different discretization strategies; in particular, it is possible to implement spectral and semi-Lagrangian methods, which allows us to design numerical schemes that are not encumbered by the sonic CFL condition.


  1. dynamical low-rank approximation
  2. projector splitting
  3. Boltzmann equation
  4. fluid dynamics
  5. weakly compressible flow

MSC codes

  1. 15A69
  2. 76M25
  3. 65M99

Get full access to this article

View all available purchase options and get full access to this article.


S. Ansumali and I. V. Karlin, Kinetic boundary conditions in the lattice Boltzmann method, Phys. Rev. E, 66 (2002), 026311.
S. Ashby et al., The Opportunities and Challenges of Exascale Computing, Report of the ASCAC Subcommittee on Exascale Computing, 2010.
C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46 (1993), pp. 667--753.
C. Bardos, F. Golse, and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), pp. 323--344.
J. B. Bell, P. Colella, and H. M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), pp. 257--283.
Y. Bo, P. Wang, Z. Guo, and L. Wang, DUGKS simulations of three-dimensional Taylor--Green vortex flow and turbulent channel flow, Comput. Fluids, 155 (2017), pp. 9--21.
S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), pp. 329--364.
D. Conte and C. Lubich, An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics, ESAIM Math. Model. Numer. Anal., 44 (2010), pp. 759--780.
L. Einkemmer and C. Lubich, A low-rank projector-splitting integrator for the Vlasov--Poisson equation, SIAM J. Sci. Comput., 40 (2018), pp. B1330--B136,
L. Einkemmer and C. Lubich, A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput., to appear.
L. Einkemmer and M. Wiesenberger, A conservative discontinuous Galerkin scheme for the 2D incompressible Navier--Stokes equations, Comput. Phys. Commun., 185 (2014), pp. 2865--2873.
A. Fakhari and T. Lee, Numerics of the lattice Boltzmann method on nonuniform grids: Standard LBM and finite-difference LBM, Comput. Fluids, 107 (2015), pp. 205--213.
Z. Guo, K. Xu, and R. Wang, Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case, Phys. Rev. E, 88 (2013), 033305.
J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B, 94 (2016), 165116.
T. Jahnke and W. Huisinga, A dynamical low-rank approach to the chemical master equation, J. Math. Biol., 70 (2008), pp. 2283--2302.
E. Kieri, C. Lubich, and H. Walach, Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal., 54 (2016), pp. 1020--1038.
J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177 (1987), pp. 133--166.
O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434--454.
O. Koch and C. Lubich, Dynamical tensor approximation, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360--2375.
K. Kormann, A semi-Lagrangian Vlasov solver in tensor train format, SIAM J. Sci. Comput., 37 (2015), pp. 613--632.
A. Krämer, K. Küllmer, D. Reith, W. Joppich, and H. Foysi, Semi-Lagrangian off-lattice Boltzmann method for weakly compressible flows, Phys. Rev. E, 95 (2017), 023305.
P. Lallemand and L. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), pp. 6546--6562.
P. Lavallee, J. P. Boon, and A. Noullez, Boundaries in lattice gas flows, Phys. D, 47 (1991), pp. 233--240.
M. Lentine, J. T. Grétarsson, and R. Fedkiw, An unconditionally stable fully conservative semi-Lagrangian method, J. Comput. Phys., 230 (2011), pp. 2857--2879.
J. Liu and C. Shu, A high-order discontinuous Galerkin method for 2D incompressible flows, J. Comput. Phys., 160 (2000), pp. 577--596.
C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, 2008.
C. Lubich, Time integration in the multiconfiguration time-dependent Hartree method of molecular quantum dynamics, Appl. Math. Res. Express. AMRX, 2015 (2015), pp. 311--328.
C. Lubich and I. V. Oseledets, A projector-splitting integrator for dynamical low-rank approximation, BIT, 54 (2014), pp. 171--188.
C. Lubich, I. V. Oseledets, and B. Vandereycken, Time integration of tensor trains, SIAM J. Numer. Anal., 53 (2015), pp. 917--941.
C. Lubich, T. Rohwedder, R. Schneider, and B. Vandereycken, Dynamical approximation by hierarchical Tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 470--494.
C. Lubich, B. Vandereycken, and H. Walach, Time Integration of Rank-Constrained Tucker Tensors, preprint, arXiv:1709.02594, 2017.
R. S. Maier, R. S. Bernard, and D. W. Grunau, Boundary conditions for the lattice Boltzmann method, Phys. Fluids, 8 (1996), pp. 1788--1801.
H. Mena, A. Ostermann, L. Pfurtscheller, and C. Piazzola, Numerical Low-Rank Approximation of Matrix Differential Equations, arXiv:1705.10175, 2017.
H.-D. Meyer, F. Gatti, and G. A. Worth, Multidimensional Quantum Dynamics, John Wiley & Sons, New York, 2009.
H. D. Meyer, U. Manthe, and L. S. Cederbaum, The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett., 165 (1990), pp. 73--78.
M. Min and T. Lee, A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows, J. Comput. Phys., 230 (2011), pp. 245--259.
E. Musharbash, F. Nobile, and T. Zhou, Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comput., 37 (2015), pp. A776--A810.
R. D. Nair, J. S. Scroggs, and F. H. M. Semazzi, A forward-trajectory global semi-Lagrangian transport scheme, J. Comput. Phys., 190 (2003), pp. 275--294.
A. Nonnenmacher and C. Lubich, Dynamical low-rank approximation: applications and numerical experiments, Math. Comput. Simul., 79 (2008), pp. 1346--1357.
J. M. Qiu, High-order mass-conservative semi-Lagrangian methods for transport problems, in Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, Handb. Numer. Anal. 17, Elsevier/North-Holland, Amsterdam, 2016, pp. 353--382.
A. Robert, A stable numerical integration scheme for the primitive meteorological equations, Atmos. Ocean, 19 (1981), pp. 35--46.
F. Schornbaum and U. Rüde, Massively parallel algorithms for the lattice Boltzmann method on nonuniform grids, SIAM J. Sci. Comput., 38 (2016), pp. C96--C126.
A. Staniforth and J. Côté, Semi-Lagrangian integration schemes for atmospheric models -- A review, Mon. Weather Rev., 119 (1991), pp. 2206--2223.
B. Van Leer, Upwind and high-resolution methods for compressible flow: From donor cell to residual-distribution schemes, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2006.
P. Wang, L.-P. Wang, and Z. Guo, Comparison of the LBE and DUGKS Methods for DNS of Decaying Homogeneous Isotropic Turbulence, preprint, arXiv:1601.03815, 2016.
P. Wang, L. Zhu, Z. Guo, and K. Xu, A comparative study of LBE and DUGKS methods for nearly incompressible flows, Commun. Comput. Phys., 17 (2015), pp. 657--681.
S. Wolfram, Cellular automaton fluids 1: Basic theory, J. Stat. Phys., 45 (1986), pp. 471--526.
D. Xiu and G. E. Karniadakis, A semi-Lagrangian high-order method for Navier--Stokes equations, J. Comput. Phys., 172 (2001), pp. 658--684.
K. Xu and X. He, Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations, J. Comput. Phys., 190 (2003), pp. 100--117.
X. Yin and J. Zhang, An improved bounce-back scheme for complex boundary conditions in lattice Boltzmann method, J. Comput. Phys., 231 (2012), pp. 4295--4303.
M. Yokokawa, K. Itakura, A. Uno, T. Ishihara, and Y. Kaneda, 16.4-Tflops direct numerical simulation of turbulence by a Fourier spectral method on the Earth Simulator, in Proceedings of the 2002 ACM/IEEE Conference on Supercomputing, 2002, pp. 1--17.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A2795 - A2814
ISSN (online): 1095-7197


Submitted: 3 May 2018
Accepted: 9 July 2019
Published online: 10 September 2019


  1. dynamical low-rank approximation
  2. projector splitting
  3. Boltzmann equation
  4. fluid dynamics
  5. weakly compressible flow

MSC codes

  1. 15A69
  2. 76M25
  3. 65M99



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account