Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients

In this paper, lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients, which can provide some more realistic models than their constant-coefficient counterparts, is derived through selecting equilibrium distribution function and adding the compensate function, appropriately. Effects and approximate value range of the free parameters, which are introduced to adjust the single relaxation time and equilibrium distribution function, are discussed in detail, as well as the impact of the lattice space step and velocity. Numerical simulations in different situations of this equation are conducted, including the propagation and interaction of the solitons, the evolution of the non-propagating soliton and the propagation of the double-pole solutions. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current lattice Boltzmann model is a satisfactory and efficient algorithm.