Compact Difference Research Articles

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Graph Summarization: Compactness Meets Efficiency

As the volume and ubiquity of graphs increase, a compact graph representation becomes essential for enabling efficient storage, transfer, and processing of graphs. Given a graph, the graph summarization problem asks for a compact representation that consists of a summary graph and the corrections, such that we can recreate the original graph from the representation exactly. Although this problem has been studied extensively, the existing works either trade summary compactness for efficiency, or vice versa. In particular, a well-known greedy method provides the most compact summary but incurs prohibitive time cost, while the state-of-the-art algorithms with practical overheads are more than 20% behind in summary compactness in our comparison with the greedy method. This paper presents Mags and Mags-DM, two algorithms that aim to bridge the compactness and efficiency in graph summarization. Mags adopts the existing greedy paradigm that provides state-of-the-art compactness, but significantly improves its efficiency with a novel algorithm design. Meanwhile, Mags-DM follows a different paradigm with practical efficiency and overcomes its limitations in compactness. Moreover, both algorithms can support parallel computing environments. We evaluate Mags and Mags-DM on graphs up to billion-scale and demonstrate that they achieve state-of-the-art in both compactness and efficiency, rather than in one of them. Compared with the method that offers state-of-the-art compactness, Mags and Mags-DM have a small difference (< 0.1% and < 2.1%) in compactness. For efficiency, Mags is on average 11.1x and 4.2x faster than the two state-of-the-art algorithms with practical overheads, while Mags-DM can further reduce the running time by 13.4x compared with Mags. This shows that graph summarization algorithms can be made practical while still offering a compact summary.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

Comprehensive analysis of colloid formation, distribution, and properties of monovarietal red wines using asymmetrical flow field-flow fractionation with online multidetection

Red wine colloids, crucial in determining wine quality and stability, are understudied due to inadequate techniques for studying them effectively in the natural wine environment. Recently, Asymmetrical Flow Field-flow Fractionation (AF4) with online multidetection has emerged as a novel analytical tool for quantifying, fractionating, and characterizing red wine colloids in their native state. This study aimed to characterize the colloidal composition of 24 monovarietal Italian wines produced without filtration, oak contact, fining treatments, malolactic fermentation, macerating enzymes or ageing on yeast lees. AF4 analysis allowed quantification and characterization of wine colloids based on light scattering signal (MALS; gyration radius − Rg), size (hydrodynamic radius – Rh) and absorbance (A280 & A520 nm).The results showed that each wine contained up to five distinct colloids’ populations, varying in size and gyration radii. Despite possessing very similar Rh, most colloids exhibited great differences in compactness, as indicated by their varying Rg values. Comparing the A280 signal of whole wines to those of wines containing only species larger than 5 kDa (considered colloids) allowed to calculate the percentage of molecules involved in colloidal particles assembly, ranging from 1 to 44 % of the total A280 absorbing compounds, reflecting the diversity among wines. The A520 signal indicated the presence of polymeric pigments in the colloidal fraction. Notably, colored colloids all had Rg > 20 nm, indicating their association with other colloidal-forming compounds. This observation led to the conclusion that, apart from free anthocyanins and polymeric pigments, the color of red wines is also due to colloidal particles formed by the latter bound to proteins, with their quantity being highly variable across wines of different origin. These findings, which highlight the fundamental role of proteins in shaping the colloidal status of red wines, were utilized to propose an updated hypothetical model for colloidal aggregation in red wine.

Accelerated schemes of compact difference methods for space‐fractional sine‐Gordon equations with distributed delay

AbstractIn this paper, for quickly solving one‐ and two‐dimensional space‐fractional sine‐Gordon equations with distributed delay, we suggest several accelerated schemes of direct compact difference (DCD) methods. For one‐dimensional (1D) problems, with a function transformation, we construct an indirect compact difference (ICD) method, which requires less calculation cost than the corresponding DCD method, and prove under the appropriate conditions that ICD method has second‐order (resp. forth‐order) calculation accuracy in time (resp. space). By extending the argument for 1D case, we further obtain an ICD method for solving two‐dimensional (2D) problems and derive the similar convergence result. For ICD and DCD methods of 2D problems, we also give their alternative direction implicit (ADI) schemes. Moreover, for the fast implementations of ICD method of 1D problems and indirect ADI method of 2D problems, we further present their acceleration strategies. Finally, with a series of numerical experiments, the findings in this paper are further confirmed.

Design of the Monopulse Feeding Network for a Slotted Waveguide Array on an Annular Disk

Design of a monopulse feeding network including a compact power distribution network and a monopulse comparator for a dual polarization slotted waveguide array fabricated on an annular disk is presented in this paper. As the slotted waveguide array is arranged on an annular disk, the feeding network is more complicated than that of a regular array such as a rectangular array. The design details of some key waveguide components, such as the compact assembly of H-plane T-junctions and E-plane elbows used to connect power distribution networks and radiation waveguides, are provided. Quasiplanar magic tees are designed and used to construct the compact sum and difference comparator. The antenna system contains two comparators, which are used to generate sum and difference beams for horizontal polarization and vertical polarization, respectively. Finally, the monopulse slotted waveguide array antenna is divided into two modules, a comparator module and an antenna module (including power distribution network), and fabricated with the layered processing and bonding process. The comparator module is measured using a network analyzer to verify its amplitude-frequency characteristics and phase-frequency characteristics. Screwing the comparator module and the antenna module together, a monopulse slotted waveguide array antenna is obtained, and the sum beam and difference beam characteristics of the antenna are measured in a microwave chamber and presented.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

Conservative compact and monotone fourth order difference schemes for quasilinear equations

Conservative compact and monotone fourth order difference schemes for quasilinear equations

Targeting Bacterial Nanocellulose Properties through Tailored Downstream Techniques.

Bacterial nanocellulose (BNC) is a biodegradable polysaccharide with unique properties that make it an attractive material for various industrial applications. This study focuses on the strain Komagataeibacter medellinensis ID13488, a strain with the ability to produce high yields of BNC under acidic growth conditions and a promising candidate to use for industrial production of BNC. We conducted a comprehensive investigation into the effects of downstream treatments on the structural and mechanical characteristics of BNC. When compared to alkaline-treated BNC, autoclave-treated BNC exhibited around 78% superior flexibility in average, while it displayed nearly 40% lower stiffness on average. An SEM analysis revealed distinct surface characteristics, indicating differences in cellulose chain compaction. FTIR spectra demonstrated increased hydrogen bonding with prolonged interaction time with alkaline solutions. A thermal analysis showed enhanced thermal stability in alkaline-treated BNC, withstanding temperatures of nearly 300 °C before commencing degradation, compared to autoclaved BNC which starts degradation around 200 °C. These findings provide valuable insights for tailoring BNC properties for specific applications, particularly in industries requiring high purity and specific mechanical characteristics.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Sixth-order compact difference scheme and multigrid method for solving the Poisson equation

Sixth-order compact difference scheme and multigrid method for solving the Poisson equation

A vertebra of a small species of Pachycetus from the North Sea and its inner structure and vascularity compared with other basilosaurid vertebrae from the same site.

In the Western Scheldt Estuary near the Belgian-Dutch border, middle to late Eocene strata crop out at the current seafloor. Most vertebrae of large Eocene basilosaurid taxa from this area were previously described in several papers. They represent three morphotypes: elongated vertebrae of a large species of Pachycetus (Morphotype 1b), a not-elongated vertebra of a large 'dorudontid' basilosaurid (Morphotype 2) and 'shortened' vertebrae of a new, unnamed taxon (Morphotype 3). This article deals with a still undescribed, smaller vertebra, NMR-16642, from this site. Our first aim was to date it by dinoflagellate cysts in adhering sediments. Yielding an age of about 38 Ma, it is one of the very few remains of basilosaurids from Europe, of which the age could be assessed with reasonable certainty. The vertebra, Morphotype 1a, is assigned to a small species of Pachycetus. High-quality CT scans are used to differentiate between NMR-16642, Morphotype 1a, and the large species of Pachycetus, Morphotype 1b. Another aim of this paper is to investigate the inner structure and vascularity of the study vertebra and that of the other morphotypes (1b, 2, 3) from this area by using high-quality CT scans. Notwithstanding differences in size, shape and compactness, the vertebral inner structure with a multi-layered cortex of periosteal bone, surrounding two cones of endosteal bone appears to be basically similar in all morphotypes. Apparently, this inner structure reflects the ontogenetic vertebral growth. An attempt to reconstruct the vascularity of the vertebrae reveals a remarkable pattern of interconnected vascular systems. From the dorsal and, if present, ventral foramina, vascular canals are running to a central vascular node. From this node a system of vascular canals goes to the epiphyseal ends, giving rise to separate systems for cortex and cones. It is the first time that the vascularity of vertebrae of archaeocetes is investigated.

The Impact of Fine-Layering of Tailings Dam on the Variation Pattern of Infiltration Lines

The type of soil and its compactness significantly influence its permeability coefficient, which in turn affects the drainage difficulty of soil pore water and the distribution of the infiltration line. However, current tailings dam models typically consider only a single soil layer instead of taking into account the differences in soil types and compactness, resulting in a deviation between simulated results and actual conditions. To address this issue, this study proposes three models with a gradually increasing degree of layering refinement based on soil type and compactness. These models aim to simulate the variations in the infiltration line under three different strategies: constant head, rainfall, and drainage. The simulation results indicate that the average increase in the infiltration line of the three schemes after rainfall is 46.2%, 65.88%, 83.52%, respectively; the fitting percentages for each scheme of infiltration line after 720 days of drainage and the constant head stage are 72.38%, 88.27%, and 93.61%, respectively. It can be seen that the higher the refinement level of the layered model, the more sensitive it is to changes in the infiltration line. Furthermore, as the refinement level of the layered model increases, the simulation effect on the changes in the infiltration line improves, and the simulated results become more consistent with the actual situation. This finding provides a strategy and possibility for the study of the tailings dam’s infiltration lines, safety, and stability.

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

<abstract><p>In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.</p></abstract>

An Efficient Compact Difference Method for the Fourth-order Nonlocal Subdiffusion Problem

An Efficient Compact Difference Method for the Fourth-order Nonlocal Subdiffusion Problem

A split iterative asymptotic method for the numerical solution of a class of fractional heat transfer equations

In this paper, a new split iterative compact difference scheme for a class of system is constructed. Then, the conservation properties of the scheme are discussed, and the convergence of the split iterative difference scheme is analyzed by using the discrete energy method on the basis of the prior estimation. Finally, numerical experiments verify these properties of the new scheme. In addition, the numerical results also show the influence of fractional derivative on the variation of the transport equation.

Numerical analysis of an uncoupled and linearized compact difference scheme with energy dissipation property for the generalized dissipative symmetric regularized long‐wave equations

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‐norm for both and . Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme.