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

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.