Acceleration of lattice Boltzmann models through state extrapolation: a reaction–diffusion example

Vandekerckhove, Christophe; Leemput, Pieter Van; Roose, Dirk

doi:10.1016/j.apnum.2007.11.018

Cited by 5 publications

(5 citation statements)

References 42 publications

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“…The focus was on the details of the coupling and how information is exchanged. The use of different ∆x and ∆t values in both regions optimized to local stability properties would be a further development, see [5,9,15,16,23,26,31,33,38] and [6,7,27,32,34,39] for guidelines. Also, we used LBM and PDE models which are equivalent in the macroscopic limit.…”

Section: Discussionmentioning

confidence: 99%

“…5.2, the first term in (38) cancels with (24). For practical computations, (37) can be substituted in the remainder of (38) to obtain an expression which is independent of ρ F D (x lbe ). The approximate equation for E(x lbe ) is then given by…”

Section: Error Using Ce-1 Coupling For Non-linear Reaction Termmentioning

confidence: 99%

“…Using different space-time grids on the sub-domains is an interesting topic of further research. For the time dependent problem, we refer to grid refinement techniques like adaptive mesh refinement (AMR) [5] for the PDE and the different approaches in [31,15,16,23] for the LBM and to certain implicit [9,26,33] or outer explicit [38] time discretization schemes. For the time independent problem, ideas from multigrid [34] for the PDE and [27,32] for the LBM or iterative methods, like the preconditioned Generalized Method of Residuals (GMRES) [6,7] or the biconjugate gradient algorithm [39] could be used.…”

Section: Introductionmentioning

confidence: 99%

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Accuracy of Hybrid Lattice Boltzmann/Finite Difference Schemes for Reaction-Diffusion Systems

Leemput¹,

Vandekerckhove²,

Vanroose³

et al. 2007

Multiscale Model. Simul.

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In this article we construct a hybrid model by spatially coupling a lattice Boltzmann model (LBM) to a finite difference discretization of the partial differential equation (PDE) for reaction-diffusion systems. Because the LBM has more variables (the particle distribution functions) than the PDE (only the particle density), we have a one-to-many mapping problem from the PDE to the LBM domain at the interface. We perform this mapping using either results from the Chapman-Enskog expansion or a point-wise iterative scheme that approximates these analytical relations numerically. Most importantly, we show that the global spatial discretization error of the hybrid model is one order less accurate than the local error made at the interface. We derive closed expressions for the spatial discretization error at steady state and verify them numerically for several examples on the one-dimensional domain.

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Section: Discussionmentioning

confidence: 99%

Section: Error Using Ce-1 Coupling For Non-linear Reaction Termmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Accuracy of Hybrid Lattice Boltzmann/Finite Difference Schemes for Reaction-Diffusion Systems

Leemput¹,

Vandekerckhove²,

Vanroose³

et al. 2007

Multiscale Model. Simul.

Self Cite

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show abstract

“…For a fixed value of k, the extrapolation step size m can be chosen such that an adequate norm of the local error estimate (per step or per unit step) is kept below a specified (relative and/or absolute) tolerance. The outline of a possible strategy for the projective method is given in [14]. There we showed how, next to using traditional local error estimation techniques such as Richardson extrapolation or using methods of different order, the local error can also be estimated by using the theoretical error formulae derived in this paper.…”

Section: Discussionmentioning

confidence: 98%

Accuracy analysis of acceleration schemes for stiff multiscale problems

Vandekerckhove

Roose

2008

Journal of Computational and Applied Mathematics

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In the context of multiscale computations, techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks (equation-free multiscale computations). The main tool is the so-called coarse-grained time-stepper, which implements an approximation of the unavailable macroscopic time-stepper using only the microscopic simulator. Several schemes were developed to accelerate the coarse-grained time-stepper, exploiting the smoothness in time of the macroscopic dynamics. To date, mainly the stability of these methods was analyzed. In this paper, we focus on their accuracy properties, mainly in the context of parabolic problems. We study the global error of the different methods, compare with explicit stiff ODE solvers, and use the theoretical results to develop more accurate variants. Our theoretical results are confirmed by various numerical experiments.

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“…In the last decade, this method has been employed to simulate mass transfer problems as with groundwater flow equation as well as solute transport equation. The diffusion equation governs groundwater flow, while advection-diffusion equation describes the solute transport [2,9,10]. It is worth noting that various lattice configurations have been developed for LBM to solve mass transfer equations, containing D1Q2 and D1Q3 for one dimensional problems, D2Q4, D2D5, and D2Q9 for two dimensional problems, in addition to, D3Q15 and D3Q19 for the three dimensions [3,[11][12][13].…”

Section: Introductionmentioning

confidence: 99%

The solution of unconfined groundwater flow with an innovative lattice Boltzmann method

et al. 2021

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Lattice Boltzmann method (LBM) has emerged as a fast, precise, and efficient numerical solution to solve differential equations. There seems to be a dearth of research regarding the solution for groundwater flow in unconfined aquifer using LBM. Accordingly, in this study, an innovative numerical solution based on LBM was introduced to solve groundwater flow in unconfined aquifers, taking into account D2Q9 scheme. The solutions obtained from the proposed LBM were compared to results stemmed from three different unconfined groundwater problems with known solutions.Both steady and transient conditions for groundwater flow were considered in simulations. It was deduced that the proposed LBM could simulate the unconfined groundwater flow satisfactorily.

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Acceleration of lattice Boltzmann models through state extrapolation: a reaction–diffusion example

Cited by 5 publications

References 42 publications

Accuracy of Hybrid Lattice Boltzmann/Finite Difference Schemes for Reaction-Diffusion Systems

Accuracy of Hybrid Lattice Boltzmann/Finite Difference Schemes for Reaction-Diffusion Systems

Accuracy analysis of acceleration schemes for stiff multiscale problems

The solution of unconfined groundwater flow with an innovative lattice Boltzmann method

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