Volume 11, Issue 5
Truncation Errors, Exact and Heuristic Stability Analysis of Two-relaxation-times Lattice Boltzmann Schemes for Anisotropic Advection-diffusion Equation

Irina Ginzburg

Commun. Comput. Phys., 11 (2012), pp. 1439-1502.

Published online: 2012-11

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  • Abstract

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the tworelaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-diffusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxationrates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill. 

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@Article{CiCP-11-1439, author = {Irina Ginzburg}, title = {Truncation Errors, Exact and Heuristic Stability Analysis of Two-relaxation-times Lattice Boltzmann Schemes for Anisotropic Advection-diffusion Equation}, journal = {Communications in Computational Physics}, year = {2012}, volume = {11}, number = {5}, pages = {1439--1502}, abstract = {

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the tworelaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-diffusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxationrates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill. 

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.211210.280611a}, url = {http://global-sci.org/intro/article_detail/cicp/7421.html} }
TY - JOUR T1 - Truncation Errors, Exact and Heuristic Stability Analysis of Two-relaxation-times Lattice Boltzmann Schemes for Anisotropic Advection-diffusion Equation AU - Irina Ginzburg JO - Communications in Computational Physics VL - 5 SP - 1439 EP - 1502 PY - 2012 DA - 2012/11 SN - 11 DO - http://dor.org/10.4208/cicp.211210.280611a UR - https://global-sci.org/intro/cicp/7421.html KW - AB -

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the tworelaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-diffusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxationrates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill. 

Irina Ginzburg. (1970). Truncation Errors, Exact and Heuristic Stability Analysis of Two-relaxation-times Lattice Boltzmann Schemes for Anisotropic Advection-diffusion Equation. Communications in Computational Physics. 11 (5). 1439-1502. doi:10.4208/cicp.211210.280611a
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