A first-order Fourier integrator for the nonlinear Schrödinger equation on 𝕋 without loss of regularity
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In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in H^γ for any initial data belonging to H^γ, for any γ >3/2. That is, up to some fixed time T, there exists some constant C=C(u_L^∞([0,T]; H^γ))>0, such that u^n-u(t_n)_H^γ(𝕋)≤ C τ, where u^n denotes the numerical solution at t_n=nτ. Moreover, the mass of the numerical solution M(u^n) verifies |M(u^n)-M(u_0)|≤ Cτ^5. In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if u_0∈ H^1(𝕋), we rigorously prove that u^n-u(t_n)_H^1(𝕋)≤ Cτ^1/2-, where C= C(u_0_H^1(𝕋))>0.
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