论文标题
对数Schrodinger方程的局部能量正则化的错误估计
Error estimates of local energy regularization for the logarithmic Schrodinger equation
论文作者
论文摘要
对数非线性已在许多部分差分方程（PDE）中用于建模各种应用中的问题。对数函数的奇异性，它在建立数学理论时引入了核心的困难，以及对PDE的PDE的数字方法，以及此类非线性的PDE的数字方法。在这里，我们将对数Schrödinger方程（LOGSE）作为原型模型。 Instead of regularizing $f(ρ)=\ln ρ$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(ρ)=ρ\ln ρ-ρ$ locally near $ρ=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic通过能量变化，Schrödinger方程（ERLOGSE）。线性收敛是根据小型正规化参数$ 0 <\ ep \ ll1 $在erlogse和logSe的解决方案之间建立的。此外，Erlogse的保守能量是四处收敛到logSe的，这显着提高了文献中正则化方法的线性收敛速率。通过使用Lie-Trotter拆卸器来解决ERLOGSE的错误估计。据报道，数值结果是为了确认我们对LER的错误估计以及TheerLogse的时间序列集成符。最后，我们的结果表明，LER的性能要比直接使对数非线性直接正规化对数非线性的表现更好。
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schrödinger equation (LogSE)as a prototype model. Instead of regularizing $f(ρ)=\ln ρ$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(ρ)=ρ\ln ρ-ρ$ locally near $ρ=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<\ep\ll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
