Asymptotic Analysis and Optimal Error estimates for Benjamin-Bona-Mahony-Burgers' Type Equations - Sudeep Kundu (2018)
© 2018 Wiley Periodicals, Inc. 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 ∞ (H j ), j = 0, 1, 2, and W 1, ∞ (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 ∞ (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 ou r theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.