Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations
Provides explicit convergence rates for low-regularity initial data in nonlinear Schrödinger equations, addressing a gap in numerical analysis for dispersive PDEs.
The authors analyze convergence rates of dispersive numerical schemes for nonlinear Schrödinger equations, proving that schemes satisfying discrete Strichartz estimates achieve polynomial convergence for Hs data with 0 < s < 1/2, whereas non-dispersive schemes yield only logarithmic decay.
This article is devoted to the analysis of the convergence rates of several nu- merical approximation schemes for linear and nonlinear Schrödinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schrödinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that can not be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for Hs(R) data if 0 < s < 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic decay rates.