Error Estimates of a Regularized Finite Difference Method for the Logarithmic Schrödinger Equation

Abstract

We present a regularized finite difference method for the logarithmic Schrödinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. ln ρ → −∞ when ρ → 0 + with ρ = |u| 2 being the density and u being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schrödinger equation (RLogSE) is proposed with a small regularization parameter 0 < ε ≪ 1 and linear convergence is established between the solutions of RLogSE and LogSE in term of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size h and time step τ as well as the small regularization parameter ε. Finally numerical results are reported to confirm our error bounds.

Regularized numerical methods for the logarithmic Schrödinger equation

“…In the multi-dimensional case, one can actually tensorize such one-dimensional solutions due to the property ln |ab| = ln |a| + ln |b|. In the case λ < 0, if a 0 = −λ > 0, then r(t) ≡ 1, which generates a moving Gausson when v = 0 and a static Gausson when v = 0 of the LogSE; and if 0 < a 0 = −λ, the function r is (time) periodic (in agreement with the absence of dispersive effects), which generates a breather of the LogSE [7]. In the case λ > 0, the large time behavior of r does not depend on its initial data, r(t) ∼ 2t √ λα 0 ln t as t → ∞ (see [17]).…”

“…and with an error O(ε 4 4+d ) in the case of the whole space Ω = R d , provided that the first two momenta of u 0 belong to L 2 (R d ) [7]. In addition, E ε (u ε ) = E(u) + O(ε).…”

“…This important feature was noticed already from the introduction of this model [13]: if u 0 is Gaussian, u(·, t) is Gaussian for all time, and solving (1.1) amounts to solving ordinary differential equations. For the convenience of the reader, we briefly recall the formulas given in [7]. In the one-dimensional case, if…”

“…with a 0 , b 0 ∈ C satisfying α 0 := Re a 0 > 0 and v ∈ R being constants, then the solution of (1.1) is given by [4,7,17] (…”

“…Very recently, a semi-implicit finite difference method was proposed and analyzed for (1.12) and thus for (1.1) [7]. As we know, there are many efficient and accurate numerical methods for the nonlinear Schrödinger equation (1.10) such as the time-splitting spectral method [8,9,12,16,18,26,27,36,39] and the conservative Crank-Nicolson finite difference (CNFD) method [6,10].…”