Error analysis of a first-order IMEX scheme for the logarithmic Schr\"odinger equation

The logarithmic Schr\"odinger equation (LogSE) has a logarithmic nonlinearity $f(u)=u\ln |u|^2$ that is not differentiable at $u=0.$ Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularising $f(u)$ as $u^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^2$ to overcome the blowup of $\ln |u|^2$ at $u=0$ has been investigated recently in literature. With the understanding of $f(0)=0,$ we analyze the non-regularized first-order Implicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the H\"older continuity of the logarithmic term, and a nonlinear Gr\"{o}nwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first one to study the direct linearized scheme for the LogSE as far as we can tell.Comment: 19 pages, 5 figure

Ultra-Low-Frequency Radio Astronomy Observations from a Selenocentric Orbit: first results of the Longjiang-2 experiment

This paper introduces the first results of observations with the Ultra-Long-Wavelength (ULW) -- Low Frequency Interferometer and Spectrometer (LFIS) on board the selenocentric satellite Longjiang-2. We present a brief description of the satellite and focus on the LFIS payload. The in-orbit commissioning confirmed a reliable operational status of the instrumentation. We also present results of a transition observation, which offers unique measurements on several novel aspects. We estimate the RFI suppression required for such a radio astronomy instrumentation at the Moon distances from Earth to be of the order of 80 dB. We analyse a method of separating Earth- and satellite-originated radio frequency interference (RFI). It is found that the RFI level at frequencies lower than a few MHz is smaller than the receiver noise floor.Comment: Accepted for publication in Experimental Astronomy; 22 pages, 11 figure

Up to fourth-order unconditionally structure-preserving parametric single-step methods for semilinear parabolic equations

We propose and analyze a class of temporal up to fourth-order unconditionally structure-preserving single-step methods for Allen–Cahn-type semilinear parabolic equations. We first revisit some up to second-order exponential time different Runge–Kutta (ETDRK) schemes, and provide unified proofs for the unconditionally maximum-principle-preserving and mass-conserving properties. Noting that the stabilized ETDRK schemes belong to a special class of parametric Runge–Kutta schemes, we introduce the stabilized integrating factor Runge–Kutta (sIFRK) formulation to construct new high-order parametric single-step methods, and propose two strategies to eliminate the exponential effects of sIFRK: (1) a recursive approximation; (2) a combination of exponential and linear functions. Together with the nonnegativity of coefficients and non-decreasing of abscissas, the resulting two families of improved stabilized integrating factor Runge–Kutta (isIFRK) schemes can unconditionally preserve the maximum-principle and conserve the mass. The order conditions, linear stability and convergence in the l∞-norm are analyzed rigorously. We demonstrate that the proposed framework, which is explicit and free of limiters or cut-off post-processing, offers a simple, practical, and effective approach to developing high-order unconditionally structure-preserving algorithms. Comparisons with traditional schemes demonstrate the necessity of developing high-order unconditionally structure-preserving schemes. A series of numerical experiments verify theoretical results of proposed isIFRK schemes.This work was supported by the Natural Science Foundation of China (No. 11901577, 11971481, 12071481), Natural Science Foundation of Hunan (No. 2020JJ5652), Defense Science Foundation of China (No. 2021-JCJQ-JJ0523), National Key R&D Program of China (No. SQ2020YFA0709803), National Key Project (No. GJXM92579) and Research Fund of National University of Defense Technology (No. ZK19-37, ZZKY-JJ-21-01)

Economic and Technical Efficiency of the Biomass Industry in China: A Network Data Envelopment Analysis Model Involving Externalities

This paper proposes the network data envelopment analysis (DEA) model accounting for negative externalities and applies it for decomposition of profit inefficiency in the biomass-agriculture circular system (Bio-AG system). A circular structure of the Bio-AG system which is different from the previously applied network structures is assumed. Since the negative externalities (i.e., pollutant emissions from the biomass industry) occur in the Bio-AG system, the property rights are taken into consideration to model the externalities-adjusted profits. Therefore, the changes in profits due to changes in the property rights (assuming no property rights, allocating property rights to agricultural sector, and allocating property rights to biomass power generation sector) are quantified. Further, the decomposition shows that the biomass power generation sector is less affected by technical inefficiency if contrasted to allocative inefficiency in terms of the profit loss. The findings suggest that the biomass power generation technology influences the profits of the biomass industry. What is more, the inefficient allocation of resources is now the key factor undermining performance of the biomass industry. Therefore, the government should adopt measures to improve the allocation of resources and prevent excessive investments or development of less efficient technologies