Fast convergence and asymptotic preserving of the general synthetic iterative scheme

Wei Su; Lianhua Zhu; Lei Wu

doi:10.1137/20M132691X

Fast convergence and asymptotic preserving of the general synthetic iterative scheme

Wei Su, Lianhua Zhu, Lei Wu

Mechanical And Aerospace Engineering

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43 Citations (Scopus)

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Abstract

Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier-Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.

Original language	English
Pages (from-to)	B1517-B1540
Number of pages	24
Journal	SIAM Journal on Scientific Computing
Volume	42
Issue number	6
DOIs	https://doi.org/10.1137/20M132691X
Publication status	Published - 14 Dec 2020

Funding

This work was supported by the Engineering and Physical Sciences Research Council (UK) under grant EP/R041938/1, and by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sk?odowska-Curie grant agreement 793007. \ast Submitted to the journal's Computational Methods in Science and Engineering section March 23, 2020; accepted for publication (in revised form) September 9, 2020; published electronically December 14, 2020. https://doi.org/10.1137/20M132691X \bfF \bfu \bfn \bfd \bfi \bfn \bfg : This work was supported by the Engineering and Physical Sciences Research Council (UK) under grant EP/R041938/1, and by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sk\lodowska-Curie grant agreement 793007. \dagger School of Engineering, The University of Edinburgh, Edinburgh EH9 3FB, UK ([email protected]). \ddagger James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK ([email protected]). \S Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China ([email protected]).

Keywords

asymptotic Navier-Stokes preserving
false/superconvergence
gas kinetic equation

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Su-etal-JSC-2020-Fast-convergence-and-asymptotic-preservingAccepted author manuscript, 1.95 MB

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title = "Fast convergence and asymptotic preserving of the general synthetic iterative scheme",

abstract = "Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier-Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.",

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author = "Wei Su and Lianhua Zhu and Lei Wu",

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AU - Zhu, Lianhua

AU - Wu, Lei

N1 - First published in SIAM Journal on Scientific Computing in Vol. 42 Issue 6, published by the Society for Industrial and Applied Mathematics (SIAM)

PY - 2020/12/14

Y1 - 2020/12/14

N2 - Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier-Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.

AB - Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier-Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.

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Fast convergence and asymptotic preserving of the general synthetic iterative scheme

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