Asymptotic Analysis and Optimal Error estimates for Benjamin-Bona-Mahony-Burgers' Type Equations

Abstract

In this article, stabilization result for the Benjamin-Bona-Mahony-Burgers' (BBM-B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in L-infinity (H-j), j = 0, 1, 2, and W-1,W-infinity (L-2)-norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM-B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in L-infinity (H-j), j = 0, 1-norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.