Coupled lattice Boltzmann method for generalized Keller–Segel chemotaxis model

In this work, an efficient and stable lattice Boltzmann method (LBM) for generalized Keller–Segel (K–S) chemotaxis model is proposed. Through the Chapman–Enskog analysis, the proposed LBM can correctly recover to the K–S model. The stability of the proposed LBM has been improved through adding correction terms in the evolution equations. Moreover, a local computational scheme for the gradient operator, which is included in the evolution equation, is developed, making the proposed LBM be implemented locally. Hence, both 2D and 3D problems with arbitrary geometries can be processed easily. In the numerical experiments, several representative chemotaxis problems are studied, including the blow up problem in square and circle domains, two-species chemotaxis blow up problem, chemotactic bacteria pattern formation in semi-solid medium in circle domain, 3D pattern formation in liquid medium, and the tumor invasion into surrounding healthy tissue. The numerical results demonstrate the high efficiency, stability and robustness of the proposed LBM. Furthermore, the capability of the proposed LBM in handling both 2D and 3D problems with complex domain is also illustrated.