A Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann model for the coupled Burgers' equations
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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 ∇θ. 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 ∇θ 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.
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