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 \theta is 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 \theta can be obtained, and particularly, the velocity u in the coupled Burgers' equations is determined by the variable \theta and its gradient term \nabla\theta. 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 $\nabla\theta$ 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.

翻译：本文提出了一种基于Cole-Hopf变换的d维耦合Burgers方程四阶多重弛豫时间格子Boltzmann（MRT-LB）模型。首先采用Cole-Hopf变换，引入中间变量θ以消除速度u=(u₁,u₂,...,u_d)上Burgers方程中的非线性对流项。此时可得到关于变量θ的扩散方程，且特别地，耦合Burgers方程中的速度u由变量θ及其梯度项∇θ共同确定。随后，针对d维变换后的扩散方程，开发了具有自然矩的通用MRT-LB模型，并给出了相应的宏观有限差分格式。在扩散尺度下，通过Maxwell迭代方法导出了所建MRT-LB模型的四阶修正方程。借助MRT-LB模型中的自由参数，不仅可获得一致的四阶修正方程，还可通过非平衡分布函数局部计算梯度项∇θ且达到四阶精度，这表明理论上d维耦合Burgers方程的MRT-LB模型可在空间上实现四阶精度。最后，通过数值模拟对MRT-LB模型进行验证，数值结果表明所提MRT-LB模型具有四阶收敛速度，与理论分析结果一致。