A high-order implicit–explicit compact finite difference method for a nonlinear time-fractional Benjamin–Bona–Mahony–Burgers equation

We propose and analyze a high-order implicit–explicit (IMEX) compact finite difference method on the temporal graded mesh and the spatial uniform mesh for solving a nonlinear time-fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation with time-fractional order. The construction of the method is based on a transformation approach, combined with piecewise quadratic Lagrange interpolation polynomials for time discretization and an effective compact finite difference method for spatial discretization. Using a modified fractional Grönwall lemma, we prove that the method is unconditionally convergent, and that for weakly singular solutions, the method has spatial fourth-order convergence, and a temporal convergence order of when the temporal mesh grading parameter is selected properly. We also prove the stability of the method using perturbation techniques. The proposed method improves the convergence order and computational efficiency of existing methods, and its implementation is very simple because only two systems of linear algebraic equations are involved at each time level. Numerical results are presented to confirm the theoretical result.