Perfectly Matched Layers and Characteristic Boundaries in Lattice Boltzmann: Accuracy vs Cost
Abstract
Artificial boundary conditions (BCs) play a ubiquitous role in numerical simulations of transport phenomena in several diverse fields, such as fluid dynamics, electromagnetism, acoustics, geophysics, and many more. They are essential for accurately capturing the behavior of physical systems whenever the simulation domain is truncated for computational efficiency purposes. Ideally, an artificial BC would allow relevant information to enter or leave the computational domain without introducing artifacts or unphysical effects. Boundary conditions designed to control spurious wave reflections are referred to as nonreflective boundary conditions (NRBCs). Another approach is given by the perfectly matched layers (PMLs), in which the computational domain is extended with multiple dampening layers, where outgoing waves are absorbed exponentially in time. In this work, the definition of PML is revised in the context of the lattice Boltzmann method. The impact of adopting different types of BCs at the edge of the dampening zone is evaluated and compared, in terms of both accuracy and computational costs. It is shown that for sufficiently large buffer zones, PMLs allow stable and accurate simulations even when using a simple zeroth-order extrapolation BC. Moreover, employing PMLs in combination with NRBCs potentially offers significant gains in accuracy at a modest computational overhead, provided the parameters of the BC are properly tuned to match the properties of the underlying fluid flow.
References
[1] , “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” Journal of Computational Physics, Vol. 114, No. 2, 1994, pp. 185–200. https://doi.org/10.1006/jcph.1994.1159
[2] , “Application of the Perfectly Matched Absorbing Layer Model to the Linear Elastodynamic Problem in Anisotropic Heterogeneous Media,” Geophysics, Vol. 66, No. 1, 2001, pp. 294–307. https://doi.org/10.1190/1.1444908
[3] , “A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations,” Communications in Computational Physics, Vol. 4, No. 4, 2008, pp. 729–796, http://global-sci.org/intro/article_detail/cicp/7814.html.
[4] , “A Perfectly Matched Layer for the Helmholtz Equation in a Semi-Infinite Strip,” Journal of Computational Physics, Vol. 201, No. 2, 2004, pp. 439–465. https://doi.org/10.1016/j.jcp.2004.06.010
[5] , “On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer,” Journal of Computational Physics, Vol. 129, No. 1, 1996, pp. 201–219. https://doi.org/10.1006/jcph.1996.0244
[6] , “On Perfectly Matched Layer as an Absorbing Boundary Condition,” Aeroacoustics Conference, AIAA Paper 1996-1664, 1996. https://doi.org/10.2514/6.1996-1664
[7] , “A Stable, Perfectly Matched Layer for Linearized Euler Equations in Unsplit Physical Variables,” Journal of Computational Physics, Vol. 173, No. 2, 2001, pp. 455–480. https://doi.org/10.1006/jcph.2001.6887
[8] , “High-Order Methods and Boundary Conditions for Simulating Subsonic Flows,” 11th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2005-2869, 2005. https://doi.org/10.2514/6.2005-2869
[9] , “Experiments with Hermite Methods for Simulating Compressible Flows: Runge-Kutta Time-Stepping and Absorbing Layers,” 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), AIAA Paper 2007-3505, 2007. https://doi.org/10.2514/6.2007-3505
[10] , “Absorbing Boundary Conditions for Nonlinear Euler and Navier–Stokes Equations Based on the Perfectly Matched Layer Technique,” Journal of Computational Physics, Vol. 227, No. 9, 2008, pp. 4398–4424. https://doi.org/10.1016/j.jcp.2008.01.010
[11] , “On the Perfectly Matched Layer for the Boltzmann-BGK Equation and its Application to Computational Aeroacoustics,” 16th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2010-3935, 2010. https://doi.org/10.2514/6.2010-3935
[12] , “Perfectly Matched Layers for the Boltzmann Equation: Stability and Sensitivity Analysis,” Journal of Computational Physics, Vol. 509, July 2024, p. 113047. https://doi.org/10.1016/j.jcp.2024.113047
[13] , “Towards Perfectly Matching Layers for Lattice Boltzmann Equation,” Computers & Mathematics with Applications, Vol. 58, No. 5, 2009, pp. 903–913. https://doi.org/10.1016/j.camwa.2009.02.013
[14] , “An Absorbing Boundary Condition for the Lattice Boltzmann Method Based on the Perfectly Matched Layer,” Computers & Fluids, Vol. 68, Sept. 2012, pp. 203–218. https://doi.org/10.1016/j.compfluid.2012.07.017
[15] , “An Absorbing Boundary Condition Based on Perfectly Matched Layer Technique Combined with Discontinuous Galerkin Boltzmann Method for Low Mach Number Flow Noise,” Journal of Theoretical and Computational Acoustics, Vol. 26, No. 4, 2018, p. 1850011. https://doi.org/10.1142/s2591728518500111
[16] , “Boundary Conditions for Direct Simulations of Compressible Viscous Flows,” Journal of Computational Physics, Vol. 101, No. 1, 1992, pp. 104–129. https://doi.org/10.1016/0021-9991(92)90046-2
[17] , “Boundary Conditions for Direct Computation of Aerodynamic Sound Generation,” AIAA Journal, Vol. 31, No. 9, 1993, pp. 1574–1582. https://doi.org/10.2514/3.11817
[18] , “Proposed Inflow/Outflow Boundary Condition for Direct Computation of Aerodynamic Sound,” AIAA Journal, Vol. 35, No. 4, 1997, pp. 740–742. https://doi.org/10.2514/2.167
[19] , “Lattice Boltzmann Approach for Near-Field Thermal Radiation,” Physical Review E, Vol. 102, No. 4, 2020. https://doi.org/10.1103/physreve.102.043308
[20] , “An Immersed Boundary-Regularised Lattice Boltzmann Method for Modelling Fluid-Structure-Acoustics Interactions Involving Large Deformation,” Physics of Fluids, Vol. 36, No. 11, 2024. https://doi.org/10.1063/5.0234280
[21] , “Characteristic Boundary Condition for Multispeed Lattice Boltzmann Model in Acoustic Problems,” Journal of Computational Physics, Vol. 490, Oct. 2023, p. 112302. https://doi.org/10.1016/j.jcp.2023.112302
[22] , “Kinetic Theory Representation of Hydrodynamics: a Way Beyond the Navier-Stokes Equation,” Journal of Fluid Mechanics, Vol. 550, Feb. 2006, pp. 413–441. https://doi.org/10.1017/S0022112005008153
[23] , “General Solution of Lattices for Cartesian Lattice Bhatanagar-Gross-Krook Models,” Physical Review E, Vol. 81, No. 3, March 2010, p. 036702. https://doi.org/10.1103/PhysRevE.81.036702
[24] , The Lattice Boltzmann Method, Springer, Cham, Switzerland, 2017. https://doi.org/10.1007/978-3-319-44649-3
[25] , The Lattice Boltzmann Equation: For Complex States of Flowing Matter, Oxford Univ. Press, U.K., 2018. https://doi.org/10.1093/oso/9780199592357.001.0001
[26] , “A Model for Collision Processes in Gases. Amplitude Processes in Charged and Neutral One-Component Systems,” Physical Review, Vol. 94, No. 3, 1954, pp. 511–525. https://doi.org/10.1103/PhysRev.94.511
[27] , The Mathematical Theory of Non-Uniform Gases, 3rd ed.,
Cambridge Mathematical Library , Cambridge Univ. Press, U.K., 1990, https://books.google.de/books?id=y2Yyy798 WzIC.[28] , “The Mathematical Structure of the Lattices of the Lattice Boltzmann Method,” Journal of Computational Science, Vol. 17, Nov. 2016, pp. 475–481. https://doi.org/10.1016/j.jocs.2016.03.002
[29] , “From the Continuous to the Lattice Boltzmann Equation: The Discretization Problem and Thermal Models,” Physical Review E, Vol. 73, No. 5, 2006. https://doi.org/10.1103/physreve.73.056702
[30] , “Optimizing Perfectly Matched Layers in Discrete Contexts,” International Journal for Numerical Methods in Engineering, Vol. 99, No. 6, 2014, pp. 410–437. https://doi.org/10.1002/nme.4690
[31] , “A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods,” Communications in Computational Physics, Vol. 33, No. 1, 2023, pp. 101–117. https://doi.org/10.4208/cicp.oa-2022-0052
[32] , “Characteristic Boundary Condition for Thermal Lattice Boltzmann Methods,” Computers & Mathematics with Applications, Vol. 157, March 2024, pp. 195–208. https://doi.org/10.1016/j.camwa.2023.12.033
[33] , “Time Dependent Boundary Conditions for Hyperbolic Systems,” Journal of Computational Physics, Vol. 68, No. 1, 1987, pp. 1–24. https://doi.org/10.1016/0021-9991(87)90041-6
[34] , “Characteristic Boundary Conditions for Direct Simulations of Turbulent Counterflow Flames,” Combustion Theory and Modelling, Vol. 9, No. 4, 2005, pp. 617–646. https://doi.org/10.1080/13647830500307378
[35] , “Two-Dimensional Characteristic Boundary Conditions for Open Boundaries in the Lattice Boltzmann Methods,” Journal of Computational Physics, Vol. 302, Dec. 2015, pp. 191–199. https://doi.org/10.1016/j.jcp.2015.08.044
[36] , “Characteristic Boundary Conditions in the Lattice Boltzmann Method for Fluid and Gas Dynamics,” Journal of Computational and Applied Mathematics, Vol. 262, May 2014, pp. 51–61. https://doi.org/10.1016/j.cam.2013.09.019
[37] , “A Non-Equilibrium Bounce-Back Boundary Condition for Thermal Multispeed LBM,” Journal of Computational Science, Vol. 53, July 2021, p. 101364. https://doi.org/10.1016/j.jocs.2021.101364
[38] , “Characteristic Boundary Conditions for Simulations of Compressible Reacting Flows with Multi-Dimensional, Viscous and Reaction Effects,” Combustion Theory and Modelling, Vol. 11, No. 2, 2007, pp. 259–286. https://doi.org/10.1080/13647830600898995
[39] , Lettuce: PyTorch-Based Lattice Boltzmann Framework, Springer International Publishing, Cham, 2021, pp. 40–55. https://doi.org/10.1007/978-3-030-90539-2_3
[40] , “Toward Learning Lattice Boltzmann Collision Operators,” European Physical Journal E, Vol. 46, No. 3, 2023, p. 10. https://doi.org/10.1140/epje/s10189-023-00267-w
