A High-Order Method with a Temporal Nonuniform Mesh for a Time-Fractional Benjamin–Bona–Mahony Equation

The solution of a time-fractional differential equation often exhibits a weak singularity near the initial time. It makes classical numerical methods with uniform mesh usually lose their accuracy. Technique of nonuniform mesh was found to be a very efficient approach in the literatures to recover the full accuracy based on reasonable regularity of the solution. In this paper, we study finite difference scheme with temporal nonuniform mesh for time-fractional Benjamin–Bona–Mahony equations with non-smooth solutions. Our approximation bases on an integral equation equivalent to the nonlinear problem under consideration. We employ high-order interpolation formulas to obtain a linearized scheme on a nonuniform mesh and, by using a modified Grönwall inequality established recently, we show that the proposed scheme with a temporal graded mesh is unconditionally third-order convergent in time with respect to discrete \(H^1\)-norm. Besides high order convergence the proposed scheme has the advantage that only linear systems are needed to be solved for obtaining approximated solutions. Numerical examples are provided to justify the accuracy.