Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping

Abstract

This work is devoted to the analysis of a fully-implicit numerical scheme for the critical generalized Korteweg–de Vries equation (GKdV with p = 4) in a bounded domain with a localized damping term. The damping is supported in a subset of the domain, so that the solutions of the continuous model issuing from small data are globally defined and exponentially decreasing in the energy space. Based in this asymptotic behavior of the solution, we introduce a finite difference scheme, which despite being one of the first order, has the good property to converge in L 4-strong. Combining this strong convergence with discrete multipliers and a contradiction argument, we show that the smallness of the initial condition leads to the uniform (with respect to the mesh size) exponential decay of the energy associated to the scheme. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.