[Paper Review] A Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann model for the coupled Burgers' equations

Ying Chen, Xi Liu+2 more|arXiv (Cornell University)|Sep 6, 2023

Lattice Boltzmann Simulation StudiesEngineering3 citations

TL;DR

This paper proposes a fourth-order accurate multiple-relaxation-time lattice Boltzmann model for d-dimensional coupled Burgers' equations by applying the Cole-Hopf transformation to eliminate nonlinear convection terms, transforming the system into a diffusion equation in terms of an auxiliary variable θ. The model achieves fourth-order spatial accuracy through a modified Maxwell iteration analysis and non-equilibrium distribution function reconstruction, validated numerically with a consistent fourth-order convergence rate across multiple test cases.

ABSTRACT

In this work, a Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for d-dimensional coupled Burgers' equations is developed. We first adopt the Cole-Hopf transformation where an intermediate variable θis introduced to eliminate the nonlinear convection terms in the Burgers' equations on the velocity u=(u_1,u_2,...,u_d). In this case, a diffusion equation on the variable θcan be obtained, and particularly, the velocity u in the coupled Burgers' equations is determined by the variable θand its gradient term ablaθ. Then we develop a general MRT-LB model with the natural moments for the d-dimensional transformed diffusion equation and present the corresponding macroscopic finite-difference scheme. At the diffusive scaling, the fourth-order modified equation of the developed MRT-LB model is derived through the Maxwell iteration method. With the aid of the free parameters in the MRT-LB model, we find that not only the consistent fourth-order modified equation can be obtained, but also the gradient term $ ablaθ$ can be calculated locally by the non-equilibrium distribution function with a fourth-order accuracy, this indicates that theoretically, the MRT-LB model for $d$-dimensional coupled Burgers' equations can achieve a fourth-order accuracy in space. Finally, some simulations are conducted to test the MRT-LB model, and the numerical results show that the proposed MRT-LB model has a fourth-order convergence rate, which is consistent with our theoretical analysis.

Motivation & Objective

• To develop a high-order, stable lattice Boltzmann model for d-dimensional coupled Burgers' equations, which are typically challenging due to nonlinear convection and coupling.
• To overcome the numerical instability and second-order accuracy limitations of existing single-relaxation-time and standard MRT-LB models for these equations.
• To leverage the Cole-Hopf transformation to convert the nonlinear system into a linear diffusion equation, enabling higher-order accuracy in the solution.
• To achieve fourth-order spatial accuracy in both the solution θ and its gradient ∇θ using a multiple-relaxation-time lattice Boltzmann framework with free relaxation parameters.
• To validate the theoretical accuracy through numerical simulations across varying grid resolutions and dimensions (d ≥ 1).

Proposed method

• Apply the Cole-Hopf transformation to the d-dimensional coupled Burgers' equations, introducing an auxiliary variable θ to decouple the nonlinear convection terms and reduce the system to a linear diffusion equation.
• Construct a multiple-relaxation-time (MRT-LB) model for the transformed diffusion equation using natural moments, enabling independent control of relaxation parameters for enhanced stability and accuracy.
• Derive the macroscopic finite-difference scheme from the MRT-LB model at the diffusive scaling, ensuring consistency with the underlying PDE.
• Perform a fourth-order modified equation analysis using the Maxwell iteration method to derive the leading-order error terms and identify conditions for fourth-order accuracy.
• Utilize the non-equilibrium distribution function to reconstruct the gradient term ∇θ with fourth-order spatial accuracy, under a specific condition on the relaxation parameters.
• Implement the model in numerical simulations for d-dimensional cases (d = 1, 2, 3, 4) with varying grid resolutions to verify convergence rates.

Experimental results

Research questions

• RQ1 Can a fourth-order accurate lattice Boltzmann model be constructed for d-dimensional coupled Burgers' equations using the Cole-Hopf transformation?
• RQ2 Does the MRT-LB framework allow for fourth-order spatial accuracy in both the solution θ and its gradient ∇θ through proper relaxation parameter tuning?
• RQ3 Can the proposed model maintain stability and achieve high-order convergence across different spatial dimensions (d ≥ 1) and grid resolutions?
• RQ4 Is the fourth-order accuracy of the model consistent across all components of the velocity field u and the auxiliary variable θ?
• RQ5 How does the non-equilibrium distribution function contribute to the fourth-order reconstruction of ∇θ in the MRT-LB framework?

Key findings

• The proposed MRT-LB model achieves a consistent fourth-order convergence rate in space for the auxiliary variable θ and all components of the velocity field u across d = 1 to 4 dimensions.
• Numerical simulations confirm the fourth-order convergence rate with convergence indices ranging from approximately 4.05 to 4.23 across different variables and grid refinements.
• The gradient term ∇θ is reconstructed with fourth-order accuracy using the non-equilibrium distribution function, provided the relaxation parameters satisfy a specific condition (Eq. 37).
• The model maintains stability and high-order accuracy across multiple spatial dimensions, demonstrating the universality of the MRT-LB approach for coupled nonlinear PDEs.
• The fourth-order modified equation analysis confirms that the leading-order truncation error is of order O(Δx⁴), validating the theoretical accuracy of the scheme.
• The convergence rate remains robust across a wide range of grid resolutions (e.g., from Δx = 0.2 to Δx = 0.025), with relative L2 errors decreasing proportionally to Δx⁴ as expected.