Fully decoupled and energy stable BDF schemes for a class of Keller-Segel equations

Abstract

The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, first-order and second-order approximations for a class of Keller-Segel equations based on the gradient flow structure were proposed in [48]. Mass conservation, positivity and energy stability were proved for the first-order scheme, whereas for the second-order scheme the energy stability was not provided. Besides, an explicit-implicit treatment is performed to a non-convex and non-concave term −χρϕ, making their decoupled system could only be solved in sequence. In this paper, we propose new BDF schemes of first-order (BDF1) and second-order accuracy (BDF2 and EsBDF2): the coupled term −χρϕ involved in two equations of ρ and ϕ is fully explicitly treated, thus the discrete schemes could be computed in parallel. For the first-order scheme (BDF1) and the second-order scheme (BDF2), the standard backward differentiation formula is applied to approximate the origin continuous equations with a regularization term added to guarantee the unconditional energy stability. For the BDF1 scheme, the discrete system is proved to be energy stable with a constant restriction for the stabilized parameter. For the EsBDF2 scheme, which is different with the standard BDF2 scheme, the differential operators are carefully handled and some regularization terms are added to provide the energy stability. Several numerical examples are presented to verify the theoretical results.