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Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics
Published on: April 12, 2019
Stability and stabilization of the lattice Boltzmann method
R A Brownlee1, A N Gorban, J Levesley
1Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom. r.brownlee@mcs.le.ac.uk
Summary
Researchers developed a stable, second-order accurate lattice Bhatnager-Gross-Krook (LBGK) method for fluid dynamics. This new approach addresses the stability-accuracy dilemma in lattice Boltzmann methods (LBM) using novel stabilization techniques.
Area of Science:
- Computational fluid dynamics
- Numerical analysis
- Kinetic theory
Background:
- The lattice Boltzmann method (LBM) faces a classical dilemma between stability and accuracy.
- The lattice Bhatnager-Gross-Krook (LBGK) method is a common LBM approach.
- Achieving second-order accuracy while maintaining stability is a key challenge.
Purpose of the Study:
- To develop a stable LBGK scheme with second-order accuracy for fluid dynamics.
- To analyze the stability and accuracy of LBGK schemes within a discrete dynamical system framework.
- To identify and mitigate instability mechanisms in LBGK.
Main Methods:
- Revisiting the stability-accuracy dilemma for LBM.
- Analyzing LBGK as a discrete dynamical system (free flight and entropic involution).
- Deriving necessary and sufficient conditions for second-order accuracy.
- Identifying instability mechanisms and proposing stabilization recipes.
- Numerical simulations: 1D shock tube and 2D flow around a square cylinder.
Main Results:
- Established conditions for second-order accuracy, requiring distributions on an invariant film.
- Identified primary mechanisms of instability.
- Developed simple stabilization methods with no artificial dissipation (up to second order) that ensure second-order accuracy.
- Introduced alternative methods with local artificial dissipation to prevent positivity loss and blowup.
- Validated schemes through simulations up to Re ≈ 20,000.
Conclusions:
- The proposed LBGK schemes offer a stable and second-order accurate solution for fluid dynamics.
- The discrete dynamical system framework provides natural stability and accuracy analysis.
- The invariant film is crucial for achieving second-order accuracy.
- Stabilization techniques effectively prevent instabilities and maintain positivity.
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