A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

In this paper, we present a fully discrete finite difference scheme with efficient convolution of artificial boundary conditions for solving the Cauchy problem associated with the one-dimensional linearized Benjamin-Bona-Mahony equation. The scheme utilizes the Padé expansion of the square root function in the complex plane to implement the fast convolution, resulting in significant reduction of computational costs involved in the time convolution process. Moreover, the introduction of a constant damping term in the governing equations allows for convergence analysis under specific conditions. The theoretical analysis is complemented by numerical examples that illustrate the performance of the proposed numerical method.