Numerical treatment of the Benjamin-Bona-Mahony equation using Alpert multiwavelets

摘要

This paper presents a numerical technique for solving the nonlinear Benjamin-Bona-Mahony equation. As a first step, we discretize the time by approximating the first-order time derivative via theta-weighted scheme. A system of ordinary differential equations is obtained and we solve this system using the wavelet Galerkin method by use of the Alpert multiwavelets. To this aim, the multiresolution analysis is used to construct the Alpert multiwavelets system and we introduce the wavelet transform matrix to decrease computational time. We use the energy method to prove that the time discrete scheme is unconditionally stable and convergent in time variable. An algorithm is proposed to achieve the desired error. Illustrative examples exhibit the efficiency of our method. The method produces accurate results and is easy to implement.