A fourth-order compact finite difference scheme of the two-dimensional convection-diffusion equation is proposed to solve groundwater pollution problems. A suitable scheme is constructed to simulate the law of movement of pollutants in the medium, which is spatially fourth-order accurate and temporally second-order accurate. The matrix form and solving methods for the linear system of equations are discussed. The theoretical analysis of unconditionally stable character of the scheme is verified by the Fourier amplification factor method. Numerical experiments are given to demonstrate the efficiency and accuracy of the scheme proposed, and these show excellent agreement with the exact solution.
The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. To get the numerical solution, the Crank-Nicolson finite difference method is constructed, which is second-order accurate in time and space. Numerical simulation shows excellent agreement with the analytical solution. The dynamic visualization of the simulating results is realized on ArcGIS platform. This work provides a quick and intuitive decision-making basis for water resources protection, especially in dealing with water pollution emergencies.
Single-cell RNA-sequencing (scRNA-seq) is a recent high-throughput technique that can measure gene expression, reveal cell heterogeneity, rare and complex cell populations, and discover cell types and their relationships. The analysis of scRNA-seq data is challenging because of transcripts sparsity, replication noise, and outlier cell populations. A gene coexpression network (GCN) analysis effectively deciphers phenotypic differences in specific states by describing gene–gene pairwise relationships. The underlying gene modules with different coexpression patterns partially bridge the gap between genotype and phenotype. This study presents a new framework called scGENA (single-cell gene coexpression network analysis) for GCN analysis based on scRNA-seq data. Although there are several methods for scRNA-seq data analysis, we aim to build an integrative pipeline for several purposes that cover primary data preprocessing, including data exploration, quality control, normalization, imputation, and dimensionality reduction of clustering as downstream of GCN analysis. To demonstrate this integrated workflow, an scRNA-seq dataset of the human diabetic pancreas with 1600 cells and 39,851 genes was implemented to perform all these processes in practice. As a result, scGENA is demonstrated to uncover interesting gene modules behind complex diseases, which reveal biological mechanisms. scGENA provides a state-of-the-art method for gene coexpression analysis for scRNA-seq data.
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