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Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations
Kai Koike
TL;DR
The paper tackles the gap between optimal formal convergence rates and weaker rigorous estimates for half-way bounce-back LBM in bounded incompressible Stokes flow. It adds Stokes and Knudsen-layer correctors to a prediction function and combines the resulting residual bounds with weighted L2-stability. The analysis recovers second-order velocity and first-order pressure convergence in the flat-channel setting.
Problem
Earlier rigorous estimates gave only an O(h1/2) velocity bound in the half-way configuration and did not establish pressure convergence, despite sharper formal predictions.
Method
The proof decomposes boundary consistency errors into macroscopic and kinetic components and absorbs them with Stokes and Knudsen-layer correctors.
Results
Second-order velocity and first-order pressure convergence are rigorously obtained for the half-way bounce-back D2Q9 BGK LBM in a bounded flat channel.
Takeaways & Limitations
The refined prediction function has sufficiently small residuals for weighted L2-stability to recover the sharp macroscopic convergence rates.
Takeaways & Limitations
Extensions to general relaxation collision operators, nonlinear Navier–Stokes equations, and generic non-half-way configurations remain future work.
Abstract
from arXiv · showhide
We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.
1 Introduction
The paper addresses the gap between sharp formal predictions and weaker rigorous error bounds for half-way bounce-back LBM in bounded incompressible flow. It refines the prediction-function analysis with Stokes and Knudsen-layer correctors to recover optimal rates.
- Motivation: LBM uses discrete-velocity kinetic models whose macroscopic quantities approximate continuum fluid equations, but its indirect formulation complicates rigorous analysis.The paper targets the Stokes equations and studies a D2Q9 BGK scheme.
- Prior rigorous analysis: Existing rigorous analyses established convergence for incompressible-flow LBM, including bounded domains with bounce-back boundaries, using prediction functions and weighted L2-stability.In periodic settings, those estimates already recover optimal velocity and pressure orders.
- Problem: Half-way bounce-back formally predicts second-order velocity and first-order pressure convergence, whereas earlier rigorous estimates gave only an O(h1/2) velocity bound and no pressure convergence.The loss arises from boundary consistency errors under diffusive time scaling.
- Problem: The paper rigorously seeks these optimal rates for the linear D2Q9 BGK method in a periodic flat channel with half-way walls.The physical walls lie half a lattice spacing from the nearest lattice layers.
- Method: The proof decomposes the leading boundary consistency error into macroscopic and kinetic parts, absorbed by Stokes correctors and Knudsen-layer correctors.The multidimensional construction requires solving Stokes and Knudsen-layer equations with carefully chosen data.
- Results: The refined prediction function, combined with weighted L2-stability, yields an O(h2) velocity error and O(h) pressure error.Its pointwise boundary residual is O(h5), and the resulting prediction-function-to-solution difference is bounded by O(h3).
2 Formulation, known results, and the main theorem
The paper formulates the D2Q9 BGK LBM with half-way bounce-back in a periodic flat channel, reviews prior stability and convergence results, and proves optimal rates through corrected prediction functions.
- 2.1 Geometry, D2Q9 velocity set, and notation: The scheme uses a nine-velocity D2Q9 lattice in a periodic flat channel, with physical walls positioned half a lattice spacing from boundary lattice layers.The half-way configuration is the boundary setting analyzed throughout the paper.
- 2.2 LBM scheme with the bounce-back rule: The LBM combines streaming, BGK relaxation, and the bounce-back rule to approximate the incompressible Stokes equations.The method evolves R9-valued distributions and reverses directions at walls through bounce-back.
- 2.3 Known results: L2-stability and a convergence theorem: Known analysis constructs a prediction function from Stokes data, estimates its consistency errors, and combines those estimates with weighted L2-stability.This framework yields optimal periodic-domain rates but is insufficiently sharp at bounded half-way-bounce-back boundaries.
- 2.3 Known results: L2-stability and a convergence theorem: The earlier bounded-domain theorem guarantees only O(h^{1/2}) velocity convergence and does not guarantee pressure convergence, despite formal and numerical evidence of O(h^2) velocity and O(h) pressure errors.The limitation persists even with careful initialization.
- 2.4 Main theorem: A third-order initialization is defined relative to the prediction function and supplies the initial-data condition required by the main convergence theorem.The paper also gives an example of such an initialization.
3 Refined prediction function
The refined prediction function is built by adding Stokes correctors for macroscopic boundary errors and Knudsen layer correctors for kinetic errors, yielding higher-order consistency.
- Prediction function and residuals: The prediction operator maps Stokes solutions into lattice distributions whose interior and boundary residuals quantify consistency with the LBM scheme.The total residual is decomposed into interior and boundary residuals.
- Stokes corrector: The first corrector solves Stokes equations with selected Dirichlet data to cancel the macroscopic component of the leading boundary residual.Its boundary data satisfy the required zero-flux admissibility condition.
- Refinement: The refined prediction function incorporates the Stokes corrector together with additional boundary-layer corrections.The paper explicitly introduces a refined prediction function after constructing the first corrector.
- Consistency estimates: The corrected residual analysis establishes high-order interior and boundary consistency estimates for the refined prediction function.The construction also uses Taylor expansions and residual estimates for the principal Knudsen layer profiles.
- Knudsen layer correctors: Knudsen layer correctors cancel the O(h3) kinetic boundary residual while preserving an interior residual of at least O(h5).Their profiles decay exponentially and are localized within O(1) lattice layers of the boundary.
4 Proof of the optimal error estimates
The proof combines refined consistency estimates with weighted L2 stability to derive optimal velocity and pressure error rates for the LBM scheme.
- Residual bounds: Proposition 4.1 provides residual estimates for the refined prediction function, including its boundary support and decay controls.The boundary residual is supported on the first and last lattice layers.
- Stability argument: Weighted L2 stability is combined with the residual bounds to control the difference between the actual LBM solution and the refined prediction function.The argument invokes the stability proposition and Duhamel’s principle.
- Initialization: A suitable third-order initialization controls the initial discrepancy between the lattice solution and the refined prediction function.The proof separately estimates the initialization error before applying stability and Duhamel’s principle.
5 Concluding remarks
The paper concludes that corrector-enhanced consistency analysis proves optimal rates in a flat channel and identifies extensions beyond the present setting.
- Main conclusion: The D2Q9 BGK method with half-way bounce-back achieves second-order velocity and first-order pressure convergence for incompressible Stokes flow in a flat channel.The rates agree with formal asymptotic analyses and numerical experiments.
- Main conclusion: Stokes correctors and Knudsen layers absorb the leading boundary consistency errors of the earlier prediction function.The refined prediction function has pointwise boundary consistency error O(h5), while the interior error is sufficiently high order.
- Extensions and scope: Generic non-half-way configurations remain outside the proved result, with their lower-order boundary errors requiring a new macroscopic–kinetic decomposition.Formal analysis predicts first-order velocity accuracy in that setting, while prior convergence analysis does not establish generic convergence.
- Extensions and scope: Extensions to general collision operators, nonlinear Navier–Stokes equations, and curved boundaries are left for future work.The collision-operator and nonlinear extensions require further analysis, while curved boundaries require curvature-aware Knudsen layers.
Statements and Declarations
The declarations report no competing interests, no generated or analyzed datasets, and use of ChatGPT during development of the work.
- Declarations: The author declares no competing interests.
- Declarations: No datasets were generated or analyzed during the study.
- Declarations: ChatGPT was used as an interactive research tool for proof strategies, calculations, literature searches, and exposition.