Unconditional and optimal error analysis of two linearized finite difference schemes for the logarithmic Schr\"odinger equation

In this paper, we propose two linearized finite difference schemes for solving the logarithmic Schr\"odinger equation (LogSE) without the need for regularization of the logarithmic term. These two schemes employ the first-order and the second-order backward difference formula, respectively, for temporal discretization of the LogSE, while using the second-order central finite difference method for spatial discretization. We overcome the singularity posed by the logarithmic nonlinearity $f(u)=u\ln|u|$ in establishing optimal $l^{2}$-error estimates for the first-order scheme, and an almost optimal $l^{2}$-error estimate for the second-order scheme. Compared to the error estimates of the LogSE in the literature, our error bounds not only greatly improve the convergence rate but also get rid of the time step restriction. Furthermore, without enhancing the regularity of the exact solution or imposing any requirements on the grid ratio, we establish error estimates of the two proposed schemes in the discrete $H^{1}$ norm. However, the existing results available in the literature either fail to provide $H^{1}$ error estimates or require certain restrictions on the grid ratio. Numerical results are reported to confirm our error estimates and demonstrate rich dynamics of the LogSE.