Compact Difference Research Articles

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

A fast compact difference scheme on graded meshes for solving the time-fractional nonlinear KdV equation

The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance.

Non-destructive compaction evaluation and comparison of different asphalt centerline longitudinal joints

Longitudinal joint quality is critical to the service life of asphalt pavements. Maintaining failed joints of asphalt pavements is a challenge for many highway agencies. Long-term performance issues of flexible pavements may result from a poorly constructed joint; the low-density joint can prematurely deteriorate an otherwise sound pavement. The longitudinal joint is built using different geometries, rolling patterns, and construction techniques, all involving risk in achieving a well-compacted joint. Conventional quality control (QC) and quality assurance (QA) testing (using cores or density gauges) provide limited coverage due to their spot-test nature. Using conventional QC/QA methods involves the risk of accepting a poorly constructed longitudinal joint at the time of construction. This study used continuous data from the non-destructive dielectric profiling system (DPS) to identify compaction ability differences between the various joint types and construction techniques. The analysis suggests avoiding constructing an unconfined joint when possible. Percent within limits (PWL) analysis shows that any joint geometry could produce minimal compaction differences between the joint and the mat. However, if an unconfined joint needs to be constructed, a portion of the unconfined joint should be cut back before paving the adjacent lane. Such a cutback technique can result in better unconfined joint compaction. Further, PWL results showed that using a smaller subsection/sublot can isolate local compaction issues; a 100 ft sublot length is preferable for PWL analysis when utilizing DPS for compaction evaluation.

Two fast finite difference methods for a class of variable-coefficient fractional diffusion equations with time delay

This paper introduces the fast Crank–Nicolson (CN) and compact difference schemes for solving the one- and two-dimensional fractional diffusion equations with time delay. The CN method is employed for temporal discretization, while the fractional centered difference (FCD) formula discretizes the Riesz space derivative. Additionally, a novel fourth-order scheme is developed using compact difference operators to improve accuracy. The convergence and stability of these schemes are rigorously proven. The discretized systems combining Toeplitz-like structures can be effectively solved by Krylov subspace solvers with suitable preconditioners. Each time level of these methods requires a computational complexity of O(plogp) per iteration and a memory of O(p), where p represents the total number of grid points in space. Numerical examples are given to illustrate both the theoretical results and the computational efficiency of the fast algorithm.

A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky equation

A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky equation

Derivation and error analysis of three-level linearized difference schemes for solving the Burgers-Fisher equation

ABSTRACT The main scope of this paper is to develop and analyse three-level linearized difference schemes for solving the classical Burgers-Fisher equation. For the Dirichlet boundary value problem, the first three-level linearized difference scheme is second-order accurate in both time and space. It is able to obey a discrete conservative law. By the discrete energy argument and induction, it is rigorously proved to be uniquely solvable and unconditionally convergent. Furthermore, with the purpose of improving the numerical accuracy in space, another three-level linearized compact difference scheme is then established together with some investigation on its discrete conservation law, unique solvability and unconditional convergence of order two in time and four in space. The coupled nonlinearity of Burgers' type and Fisher type is intensively treated via the order of reduction and Taylor expansion. In addition, extensions to the problem subject to the periodic boundary condition (PBC) are involved. To the best of our knowledge, this is a rare work to carefully design and rigorously analyse the high-order linearized difference schemes for solving the Burgers-Fisher equation. Numerical tests are provided to support the theoretical results and to verify the computational efficiency.

2D acoustic equation prestack reverse-time migration based on an optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite-difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective RTM method suitable for coarse grid conditions.

Feasibility of Induced Magnetic Gradient Surveying for Seepage Detection in Earth-filled Dams: Insights from Synthetic and Field Studies

Seepage is a common hydrogeological hazard in engineering. Determining the seepage paths is vital for de-risking the instability of embankment structures. With the improvement of the acquisition accuracy of magnetic sensors, the magnetometric resistivity method has become an emerging technology for detecting seepage paths through earth-filled dams. This technique is non-destructive and gives prominent signals. However, the resulting magnetic data have seen ambiguity in fully determining the targets. We propose an induced magnetic gradient surveying approach to monitor seepage paths in earth-filled dams. First, we briefly review the electromagnetic theory for the magnetic gradient tensor based on Maxwell’s equations. To match against the measurements, we present an accurate modeling framework using the third-order finite element method and a novel compact difference scheme. We verify our approach on both semi-analytical 1D and 3D models. Systematic modeling studies are then carried out to investigate the spatial distribution characteristics and sensitivities of the induced magnetic gradient to the seepage in typical dam scenarios. In addition, we conducted two field experiments in the Zhongmou experimental base and Xixiayuan Reservoir in Henan Province, China,respectively. The induced magnetic field vector and its gradient components were both acquired. Cross-validation with a-priori geological information shows that the seepage path can be spatially identified by the induced magnetic gradient components Byy, Byz, Bzy, and Bzz while the field components failed to locate the seepage pathways. This successful application indicates that the proposed approach could be a promising solution for seepage path discrimination in earth-filled dams with high resolution.

Integrating cover crops and organic amendments to mitigate the limitations of tillage on soil health and cotton productivity

AbstractVarious tillage systems have limitations on soil health, such as the degradation of soil structure and organic matter under conventional tillage (CT) systems, as well as short‐term soil compaction in conservation tillage systems. A 3‐year field experiment was established to evaluate the integration of cover crop (CC) and organic amendments (OAs) into CT and strip tillage (ST) systems, and their impact on soil properties and cotton (Gossypium hirsutum L.) productivity. The CC was cereal rye (Secale cereale), and the combined application of animal manure and biochar constituted the OA. In the third year, differences in soil compaction between the CT and ST systems were observed when the measurements were made after tillage. Moreover, integrating CC and OA under the CT and ST systems increased the soil depth to compaction zones. Soil compaction was observed at 27.5‐cm depth under CT, at 30‐cm depth under CT integrated with CC and OA, at 10‐cm depth under ST, and at 15‐cm depth under ST integrated with CC and OA, using 2 MPa as the threshold. In general, the integration of CC and OA tended to increase soil respiration, organic matter, and available nutrients, but the effects were not consistent across years and soil depth. Despite differences in the various soil health properties, the management systems had minimum impact on cotton productivity and fiber quality, indicating the ST was effective in preparing the seedbed. Moreover, the differences in soil properties were not at yield‐limiting levels within 3 years of the study.

Reflective imaging of myelin integrity in the human and mouse central nervous systems.

The structural integrity of myelin sheaths in the central nervous system (CNS) is crucial for the maintenance of its function. Electron microscopy (EM) is the gold standard for visualizing individual myelin sheaths. However, the tissue processing involved can induce artifacts such as shearing of myelin, which can be difficult to distinguish from true myelin abnormalities. Spectral confocal reflectance (SCoRe) microscopy is an imaging technique that leverages the differential refractive indices of compacted CNS myelin in comparison to surrounding parenchyma to detect individual compact myelin internodes with reflected light, positioning SCoRe as a possible complementary method to EM to assess myelin integrity. Whether SCoRe is sensitive enough to detect losses in myelin compaction when myelin quantity is otherwise unaffected has not yet been directly tested. Here, we assess the capacity of SCoRe to detect differences in myelin compaction in two mouse models that exhibit a loss of myelin compaction without demyelination: microglia-deficient mice (Csf1r-FIRE Δ/Δ) and wild-type mice fed with the CSF1R inhibitor PLX5622. In addition, we compare the ability to detect compact myelin sheaths using SCoRe in fixed-frozen versus paraffin-embedded mouse tissue. Finally, we show that SCoRe can successfully detect individual sheaths in aged human paraffin-embedded samples of deep white matter regions. As such, we find SCoRe to be an attractive technique to investigate myelin integrity, with sufficient sensitivity to detect myelin ultrastructural abnormalities and the ability to perform equally well in tissue preserved using different methods.

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

The consequences of calcium: investigating intracortical reproductive signals in the American alligator for sex determination.

Identifying sex in extinct archosaurs has proven difficult due, in part, to low sample sizes, preservation biases, and methodology. While previous studies have largely focused on morphological traits, here we investigate intracortical signals of egg-shelling in extant alligators. Egg-shelling requires large mobilizations of calcium reserves. Aves utilize medullary tissue as a calcium reserve, whereas crocodylians mobilize calcium from cortical bone or osteoderms. If crocodylians derive calcium from bone cortices for egg-shelling, then egg-shelling events should be detectable in female crocodylian cortical bone. We examined mid-diaphyseal Alligator mississippiensis femoral bone cross-sections for signals of reproduction. Compaction and area of resorbed tissue were measured in femoral cross-sections from captive raised male (n = 10) and female (n = 29) A. mississippiensis of 26-27 years at age of death. This sample is more robust than previous studies, though reproductive history data is unknown. Femora from a small sample of wild caught male (n = 6) and female (n = 6) A. mississippiensis were also measured. Data were analyzed by pairwise t-tests between sex and captivity status. There was no significant difference in either compaction or resorbed tissue values between male and female alligators, regardless of habitat (wild or captive-raised). A reproductive signal was undetectable in this study and any quantifiable differences between sexes appears to be driven by size dimorphism. Cortical resorption rates in the femora of male and female alligators are reflective of normal aging processes and not indicative of egg-shelling during reproduction. Examination of younger alligators would clarify processes driving bone turnover during reproductively active years.

Compactness of Native Vowel Categories in Monolingual, Bilingual, and Multilingual Speakers: Is Category Compactness Affected by the Number of Languages Spoken?

Phonetic category compactness pertains to the degree of variation or dispersion within a specific category. Previous research has shown that more compact native (L1) categories in production have been related to the discrimination of non-native sounds in perception and production. The understanding of the factors influencing L1 category compactness remains limited. Some proposals suggest that compactness may be influenced by individual differences in cognitive processes. Alternatively, category compactness could be linked to linguistic factors, such as the number of languages spoken or the density of the phonological system. This study investigates the latter hypothesis. This study examined category compactness in perception for three L1 Spanish vowels /i/, /e/, and /a/ across four participant groups: 12 monolinguals, 31 functional monolinguals, 24 bilinguals, and 19 multilinguals. To measure compactness in perception, the study employed a perceptual categorization task consisting of synthesized variants of /i/, /e/, and /a/. Participants were asked to label these variants as either acceptable or unacceptable members of their L1 /i/, /e/, and /a/ categories. The findings revealed significant differences in category compactness between monolingual and bi/multilingual speakers. More specifically, bilingual and multilingual speakers had larger/less compact L1 vowel categories than monolinguals. The substantial variability in compactness across all groups suggests that compactness may be influenced by a range of other individual differences, besides the number of languages spoken.

Competition and drought affect cleistogamy in a non-additive way in the annual ruderal Lamium amplexicaule.

Competition affects mixed-mating strategies by limiting available abiotic or biotic resources such as nutrients, water, space, or pollinators. Cleistogamous species produce closed (cleistogamous, CL), obligately selfed, simultaneously with open (chasmogamous, CH), potentially outcrossed flowers. The effects of intraspecific competition on fitness and cleistogamy variation can range from limiting the production of costly CH flowers because of resource limitation, to favouring CH production because of fitness advantages of outcrossed, CH offspring. Moreover, the effects of competition can be altered when it co-occurs with other environmental variations. We grew plants from seven populations of the ruderal Lamium amplexicaule, originating from different climates and habitats, in a common garden experiment combining drought, interspecific competition, and seasonal variation. All these parameters have been shown to influence the degree of cleistogamy in the species on their own. In spring, competition and drought negatively impacted fitness, but the CL proportion only increased when plants were exposed to both treatments combined. We did not observe the same results in autumn, which can be due to non-adaptive phenotypic variation, or to differences in soil compactness between seasons. The observed responses are largely due to phenotypic plasticity, but we also observed phenotypic differentiation between populations for morphological, phenological, and cleistogamy traits, pointing to the existence of different ecotypes. Our data do not support the hypothesis that CL proportion should decrease when resources are scarce, as plants with reduced growth had relatively low CL proportions. We propose that variation in cleistogamy could be an adaptation to pollinator abundance, or to environment-dependent fitness differences between offspring of selfed and outcrossed seeds, two hypotheses worth further investigation. This opens exciting new possibilities for the study of the maintenance of mixed-mating systems using cleistogamous species as models that combine the effects of inbreeding and reproductive costs.

The influence of stress on the fracture and elastic properties of carbonate rocks controlled by strike-slip faults: a novel rock-physics modelling perspective

SUMMARY The fine-scale fractures within strike-slip faults substantially impact the flowing capacity. However, effective methods for their characterization are still lacking, making it challenging to predict hydrocarbon accumulation patterns. In this study, we conducted microscopic statistics, ultrasonic experiments and theoretical modelling to analyse the fracture density and elastic characteristics within the strike-slip fault and investigated the impact of stress. Our findings reveal that the fracture density in the fault core is 3–4 times higher than that in the damage zone, and the acoustic velocity is 13–18 per cent lower under atmospheric pressure. With the rising confining pressure, the fracture density initially decreases rapidly and then slowly, while the acoustic velocity follows the same increasing trend. The gradually slowing trend indicates that the majority of fractures close within the range of 0–20 MPa. Moreover, the stress sensitivity of the bulk modulus is higher than that of the shear modulus. The stress sensitivity is higher in the fault core than in the damage zone, which correlates strongly with the variation in fracture density. These indicate that the stress sensitivity in the fault-controlled rock is attributed to stress-induced fracture deformation, predominantly manifested as volumetric compression deformation. During the geological evolution, differences in tectonic faulting, fluid filling and compaction within the fault zone contribute to present heterogeneity in fracture density. Finally, our research demonstrates a strong correlation between theoretical prediction results and underground logging, drilling and core data. These findings can help predict the underground fracture distribution and elastic response of carbonate reservoirs controlled by strike-slip faults.

Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations

In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.