Temporal Accuracies Research Articles

Analysis of the impact of high temporal resolution tropospheric parameters on GNSS station coordinate and troposphere estimation

Abstract Troposphere modelling accuracy can seriously influence the performance of precise global navigation satellite system (GNSS) positioning. The temporal interpolation accuracies of troposphere parameters become important with the improvement in troposphere modelling, especially during the periods of severe weather changes. In this study, a one-month experiment was conducted for 10 EUREF Permanent GNSS Network stations to analyse the temporal interpolation errors of numerical weather model derived troposphere parameters and their influences on GNSS station coordinate and troposphere estimation. The root mean square of the differences between the 1- and 6-hourly European Centre for Medium-Range Weather Forecasts reanalysis v5-derived troposphere parameters are 0.72/11.52 mm for the zenith hydrostatic/zenith wet delay (ZWD), while they are 7.09/91.84 mm for the slant hydrostatic/wet delay at a 3 ∘ elevation angle. The corresponding influences on the estimated upper component and ZWD are 1.28 and 0.79 mm, respectively, in the daily static Precise Point Positioning ambiguity resolution solutions. However, the accuracy improvements are insignificant in the daily static solutions, mainly due to the drop of temporal interpolation errors with increasing elevation angles and the influence of other unmodeled systematic errors. In the 4-hourly static solutions, the accuracies of the up component improve 0.51 mm (5.9% in relative) and 0.29 mm (3.5% in relative) during the severe and stable periods, respectively. These impacts cannot be ignored in some GNSS subdaily applications, for example, monitoring non-tidal ocean loading effects, bedrock and monument thermal elastic effects, etc.

An ensemble method for monitoring land cover changes in urban areas using dense Landsat time series data

This study proposes a new method for monitoring land cover change in urban areas using all available Landsat time series data, named the Ensemble of Bidirectional Time Series Analysis (EBTSA). In this method, the bidirectional Continuous Change Detection and Classification (CCDC) and the Chow Test are combined to improve the robustness against data scarcity in earlier times and reduce break detection errors and refine classification results. There are three key stages in this method: break detection using bidirectional CCDs, break refinement using the Chow Test, and bidirectional model integration and classification. The EBTSA method was evaluated over the Tianjin metropolitan area in China using Landsat data from 1989 to 2018. Results show that the proposed method improved spatial and temporal accuracies of both land cover classification and change detection, by reducing the influence of sparser Landsat data in the earlier years and the break detection errors. Using the land cover change results in the Tianjin area obtained using the EBTSA method, we analyzed the spatio-temporal distribution of land cover classification and change detection. It is found from the results that the Tianjin area experienced dramatic urban land changes, characterized by rapid urban expansion and noticeable transition from vegetation to urban land, with diverse changes from urban land to various nonurban land cover types. Results of this study showcase the effectiveness of the proposed method in land cover change monitoring in urban areas, which facilitates a more comprehensive understanding of urban land change dynamics and diversity.

Building a small fire database for Sub-Saharan Africa from Sentinel-2 high-resolution images

Coarse resolution sensors are not very sensitive at detecting small fire patches, making current estimations of global burned areas (BA) very conservative. Using medium or high-resolution sensors to generate BA products becomes then a priority, particularly in areas where fires tend to be small and frequent.Building on previous work that developed a small fire dataset (SFD) for Sub-Saharan Africa for 2016, this paper presents a new version of the dataset for 2019 using the two Sentinel-2 satellites (A and B) and VIIRS active fires. Total estimated BA was 4.8 Mkm2. This value was much higher than estimations from two global, coarser-spatial resolution BA products based on MODIS data for the same area and period: 80 % greater than estimates from FireCCI51 (based on MODIS 250 m bands) and 120 % larger than MCD64A1 (based on MODIS 500 m bands). The main differences were observed in those months with higher fire occurrence (November to January for the Northern Hemisphere regions and June to September for the Southern Hemisphere ones).Accuracy assessment of the SFD product was based on a novel sampling strategy designed to obtain independent fire reference perimeters. Validation results showed remarkable high accuracy values comparing to existing global BA products. Overall omission errors (OE) were estimated as 8.5 %, commission errors (CE) as 15.0 %, with a Dice Coefficient of 87.7 %. All of these estimations implied significant improvements over the global, coarser spatial resolution BA products (OE > 50 % and CE > 20 % for the same area and period), as well as over the previous SFD product for 2016 of the same area, generated from a single Sentinel-2 satellite and MODIS active fires (OE = 26.5 % and CE = 19.3 %). Temporal accuracies greatly increased as well with the new product, with 92.5 % of fires detected within the first 10 days of occurrence.

Monolithic projection-based method with staggered time discretization for solving non-Oberbeck–Boussinesq natural convection flows

This paper presents an efficient monolithic projection-based method with staggered time discretization (MPM-STD) to examine the non-Oberbeck–Boussinesq (NOB) effects in several natural convection problems involving dramatic temperature-dependent changes in fluid properties. The proposed approach employs the Crank–Nicolson scheme along with staggered time discretization to discretize the momentum and energy equations. The momentum and energy equations are decoupled by evaluating the velocity vector at integral time levels (n+1), and the scalar variables (pressure and temperature) at half-integral time levels (n+12). The observed density variations in all terms result in a variable-coefficient Poisson equation, which is difficult to solve efficiently. The convergence is accelerated via adoption of an appropriate pressure-correction scheme that transforms the aforementioned Poisson equation to a constant-coefficient form. The numerical simulations concerning two-dimensional (2D) periodic NOB Rayleigh–Bénard convection (RBC) in glycerol and 2D differentially heated cavity (DHC) problem in air confirmed the second-order temporal and spatial accuracies of the proposed method. By simulating the 2D DHC problem in air and the RBC problem in liquid (water or glycerol) considering NOB effects, it is concluded that the proposed MPM-STD significantly mitigates the time-step restriction, thereby increasing the computational efficiency, which exceeds that of existing semi-implicit and explicit schemes. Moreover, the potential of the proposed approach with regard to solving challenging three-dimensional (3D) turbulent problems is demonstrated by performing direct simulations of turbulent RBCs under NOB effects involving temperature differences up to 60 K with a corresponding Rayleigh number (Ra=106).

An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework

In this study, we propose an efficient algorithm for solving one-dimensional coupled viscous Burgers’ equations. One of the main accomplishments of this study is to develop a stable high-order algorithm for the system of reaction–diffusion equations. The algorithm is “robust” because it is designed to prevent non-physical oscillations through an iteration procedure of a block Gauss-Seidel type. The other is to develop an efficient algorithm for the Cauchy problem. For this, we first find half of the upstream points by adopting a multi-step qth-order (q=2,3) error correction method. The algorithm is also “economical” in the sense that an interpolation strategy for finding the remaining upstream points is designed to dramatically reduce the high computational cost for solving the nonlinear Cauchy problem without damage to the order of accuracy. Three benchmark problems are simulated to investigate the accuracy and the superiority of the proposed method. It turns out that the proposed method numerically has the qth-order temporal and 4th-order spatial accuracies. In addition, the numerical experiments show that the proposed method is superior to the compared methods in the sense of the computational cost.

Evaluating the Accuracy of the Azure Kinect and Kinect v2.

The Azure Kinect represents the latest generation of Microsoft Kinect depth cameras. Of interest in this article is the depth and spatial accuracy of the Azure Kinect and how it compares to its predecessor, the Kinect v2. In one experiment, the two sensors are used to capture a planar whiteboard at 15 locations in a grid pattern with laser scanner data serving as ground truth. A set of histograms reveals the temporal-based random depth error inherent in each Kinect. Additionally, a two-dimensional cone of accuracy illustrates the systematic spatial error. At distances greater than 2.5 m, we find the Azure Kinect to have improved accuracy in both spatial and temporal domains as compared to the Kinect v2, while for distances less than 2.5 m, the spatial and temporal accuracies were found to be comparable. In another experiment, we compare the distribution of random depth error between each Kinect sensor by capturing a flat wall across the field of view in horizontal and vertical directions. We find the Azure Kinect to have improved temporal accuracy over the Kinect v2 in the range of 2.5 to 3.5 m for measurements close to the optical axis. The results indicate that the Azure Kinect is a suitable substitute for Kinect v2 in 3D scanning applications.

Time–space domain scalar wave modeling by a novel hybrid staggered-grid finite-difference method with high temporal and spatial accuracies

Staggered-grid finite-difference (SFD) scheme is favored in wave equation simulation due to its superior accuracy and stability to center-grid finite-difference (CFD) scheme. However, for the scalar wave equation (SWE) modeling, the conventional SFD (CSFD) scheme only reaches second-order accuracy in space and time for the discrete SWE; the recently developed modified SFD (MSFD) scheme improves the accuracy to (2N)th-order (N<4) but is costly because the MSFD scheme is coined by adding many extra grid points to the CSFD scheme. To tackle these issues, we develop a cost-effective hybrid SFD (HSFD) scheme, which combines the features of the CSFD and MSFD schemes; we prove that the new HSFD scheme can simultaneously reach (2N)th-order accuracy in space and time for the discrete wave equation. In addition, to deal with the optimization difficulties due to the nonlinear dispersion relation of the SFD schemes, we propose a two-step linear optimization method to improve the accuracy of the new HSFD scheme. The analyses on dispersion, stability properties and numerical simulation examples demonstrate that the new HSFD scheme owns better accuracy and stability than the CSFD scheme. Computational cost analysis shows that the HSFD scheme can be more efficient than the MSFD scheme because it only requires half the additional grid points.

Elastic Wave Modeling With High-Order Temporal and Spatial Accuracies by a Selectively Modified and Linearly Optimized Staggered-Grid Finite-Difference Scheme

High-order staggered-grid finite-difference (SFD) schemes are preferred in elastic wave simulation for geophysical problems because they decrease the accumulation of error from grid dispersion. However, most SFD approaches reach high-order spatial but limited temporal accuracy. To tackle the issue, we develop a novel temporal and spatial high-accuracy elastic SFD scheme by selectively modifying the spatial operators of the original SFD stencil. This modification has three main advantages. First, it facilitates the design of a new SFD stencil with temporal and spatial accuracies to arbitrary even-order by a Taylor-series expansion method. Second, it helps boost the accuracy further by implementing a linear optimization method. Third, the new selectively modified SFD (SMSFD) stencil needs fewer float-point operations (FPOs) than the existing temporal high-order SFD stencil. We compare our new SMSFD scheme with spatial high-order and temporal–spatial high-order SFD schemes and show that our new elastic SMSFD scheme possesses better accuracy and stability and requires fewer FPOs than these methods.

An experimental study on monitoring the phreatic line of an embankment dam based on temperature detection by OFDR

In this study, temperature detection with distributed fibers was used to obtain data for a laboratory model of seepage heat in an embankment dam, and the temperature was monitored using optical frequency domain reflection (OFDR) technology with high temporal and spatial accuracies. First, the temperature field in the infiltrated region of the dam body during the seepage test showed changes similar to that measured by OFDR temperature-sensing fibers. Then the heat transfer characteristics analysis within the embankment dam during seepage was conducted using numerical simulations of heat-flow coupling. Finally, temperature distribution characteristics during steady-state seepage for different water levels were measured, and the relationship between the temperature curve and the phreatic line was obtained. The results demonstrate that accurate temperature detection by OFDR and its high spatial resolution can reflect the characteristics of the temperature field in the seepage model of an embankment dam.

Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution Equations

Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $$0<\alpha <1$$ , and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $$1<\alpha <2$$ . By a new duality technique, pointwise-in-time error estimates of first-order and $$ (3-\alpha ) $$ -order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.

Differences in cortical activation patterns during action observation, action execution, and interpersonal synchrony between children with or without autism spectrum disorder (ASD): An fNIRS pilot study.

Engaging in socially embedded actions such as imitation and interpersonal synchrony facilitates relationships with peers and caregivers. Imitation and interpersonal synchrony impairments of children with Autism Spectrum Disorder (ASD) might contribute to their difficulties in connecting and learning from others. Previous fMRI studies investigated cortical activation in children with ASD during finger/hand movement imitation; however, we do not know whether these findings generalize to naturalistic face-to-face imitation/interpersonal synchrony tasks. Using functional near infrared spectroscopy (fNIRS), the current study assessed the cortical activation of children with and without ASD during a face-to-face interpersonal synchrony task. Fourteen children with ASD and 17 typically developing (TD) children completed three conditions: a) Watch-observed an adult clean up blocks; b) Do-cleaned up the blocks on their own; and c) Together-synchronized their block clean up actions to that of an adult. Children with ASD showed lower spatial and temporal synchrony accuracies but intact motor accuracy during the Together/interpersonal synchrony condition. In terms of cortical activation, children with ASD had hypoactivation in the middle and inferior frontal gyri (MIFG) as well as middle and superior temporal gyri (MSTG) while showing hyperactivation in the inferior parietal cortices/lobule (IPL) compared to the TD children. During the Together condition, the TD children showed bilaterally symmetrical activation whereas children with ASD showed more left-lateralized activation over MIFG and right-lateralized activation over MSTG. Additionally, using ADOS scores, in children with ASD greater social affect impairment was associated with lower activation in the left MIFG and more repetitive behavior impairment was associated with greater activation over bilateral MSTG. In children with ASD better communication performance on the VABS was associated with greater MIFG and/or MSTG activation. We identified objective neural biomarkers that could be utilized as outcome predictors or treatment response indicators in future intervention studies.

An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows

In this paper, an explicit low-storage simplified M — stage Runge-Kutta (SRK) scheme for high Reynolds-number incompressible flows is presented. In the SRK scheme, the Poisson equation is solved only once in the final substage of each time step. By taking advantage of the SRK scheme and the advanced hybrid MPI+MPI model, we have developed an efficient parallel solver for buoyancy-driven turbulent flow. The spatial and temporal accuracies of the solver are validated with Taylor-Green vortex flow. Both the RK and SRK schemes are implemented for the simulation of turbulent Rayleigh-Benard convection as well as Rayleigh-Taylor flow. The results show that the SRK scheme can save approximately 20% of the computation time.

New local radial point interpolation-FD methods for solving fractional diffusion and damped-wave problems

A new approach is proposed for developing numerical schemes for the solution of fractional diffusion and diffusion-wave equations. The novelty lies in the construction of analytical shape functions in a three-point local radial point interpolation method. The gain is that numerical computations of matrix entries are not required and as in the finite difference setting, the availability of matrix coefficients in closed-form facilitates stability and convergence analyses. For the fractional diffusion equation, a weak-form formulation is used to develop an implicit meshless finite difference method. The unconditional stability and convergence of the scheme is theoretically established and numerical examples are given to illustrate its second-order accuracy in space. The expected time convergence and the dependence of the convergence on the multiquadric shape parameter are also investigated. The technique is applied to develop numerical schemes for the fractional diffusion-wave equation. For this problem, the new scheme is shown to perform well in terms of both temporal and spatial accuracies when compared to some existing methods.

Time-space-domain temporal high-order staggered-grid finite-difference schemes by combining orthogonality and pyramid stencils for 3D elastic-wave propagation

The presently available staggered-grid finite-difference (SGFD) schemes for the 3D first-order elastic-wave equation can only achieve high-order spatial accuracy, but they exhibit second-order temporal accuracy. Therefore, the commonly used SGFD methods may suffer from visible temporal dispersion and even instability when relatively large time steps are involved. To increase the temporal accuracy and stability, we have developed a novel time-space-domain high-order SGFD stencil, characterized by ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracies ([Formula: see text]), to numerically solve the 3D first-order elastic-wave equation. The core idea of this new stencil is to use a double-pyramid stencil with an operator length parameter [Formula: see text] together with the conventional second-order SGFD to approximate the temporal derivatives. At the same time, the spatial derivatives are discretized by the orthogonality stencil with an operator length parameter [Formula: see text]. We derive the time-space-domain dispersion relation of this new stencil and determine finite-difference (FD) coefficients using the Taylor-series expansion. In addition, we further optimize the spatial FD coefficients by using a least-squares (LS) algorithm to minimize the time-space-domain dispersion relation. To create accurate and reasonable P-, S-, and converted wavefields, we introduce the 3D wavefield-separation technique into our temporal high-order SGFD schemes. The decoupled P- and S-wavefields are extrapolated by using the P- and S-wave dispersion-relation-based FD coefficients, respectively. Moreover, we design an adaptive variable-length operator scheme, including operators [Formula: see text] and [Formula: see text], to reduce the extra computational cost arising from adopting this new stencil. Dispersion and stability analyses indicate that our new methods have higher accuracy and better stability than the conventional ones. Using several 3D modeling examples, we demonstrate that our SGFD schemes can yield greater temporal accuracy on the premise of guaranteeing high-order spatial accuracy. Through effectively combining our new stencil, LS-based optimization, large time step, variable-length operator, and graphic processing unit, the computational efficiency can be significantly improved for the 3D case.

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

Fully-resolved simulations of particle-laden viscoelastic fluids using an immersed boundary method

This study reports the development of a direct simulation code for solid spheres moving through viscoelastic fluids with a range of different rheological behaviors. The numerical algorithm was implemented on an open-source finite-volume solver coupled with an immersed boundary method, and is able to perform fully-resolved simulations, wherein all flow scales associated with the particle motion are resolved. The formulation employed exploits the log-conformation tensor to avoid high Weissenberg number issues when calculating the polymeric extra stress. A number of benchmark flows were simulated using this method, to assess the accuracy of the newly-developed solver. First, the sedimentation of a sphere in a bounded domain surrounded by either Newtonian or viscoelastic fluid was computed, and the numerical results were verified by comparison with experimental and computational data from the literature. Additionally, the spatial and temporal accuracies of the algorithm were evaluated, and different transient and advection discretization schemes were investigated. Second, the rotation of a sphere in a homogeneous shear flow was studied, and again the numerical results obtained were compared to those from the literature. Good agreement is obtained for the variation in the particle rotation rate as a function of Weissenberg number, using both the newly implemented algorithm and an alternative fixed-mesh approach. Finally, the cross-stream migration of a neutrally buoyant sphere in a steady Poiseuille flow, consisting of either a Newtonian or viscoelastic suspending fluid was investigated. For the Newtonian fluid good agreement was obtained for the particle equilibrium position when compared to the well known Segré–Silberberg effect, and for the viscoelastic fluid the effect of the retardation ratio on the final particle equilibrium position was studied. Additionally, the newly-developed solver capabilities were tested to study the shear-induced particle alignment in wall-bounded Newtonian and viscoelastic fluids. The role of the fluid rheology and finite gap size on both the rate and approach pathways of the solid particles is illustrated.

Detecting modular brain states in rest and task.

The human brain is a dynamic networked system that continually reconfigures its functional connectivity patterns over time. Thus, developing approaches able to adequately detect fast brain dynamics is critical. Of particular interest are the methods that analyze the modular structure of brain networks, that is, the presence of clusters of regions that are densely interconnected. In this paper, we propose a novel framework to identify fast modular states that dynamically fluctuate over time during rest and task. We started by demonstrating the feasibility and relevance of this framework using simulated data. Compared with other methods, our algorithm was able to identify the simulated networks with high temporal and spatial accuracies. We further applied the proposed framework on MEG data recorded during a finger movement task, identifying modular states linking somatosensory and primary motor regions. The algorithm was also performed on dense-EEG data recorded during a picture naming task, revealing the subsecond transition between several modular states that relate to visual processing, semantic processing, and language. Next, we tested our method on a dataset of resting-state dense-EEG signals recorded from 124 patients with Parkinson’s disease. Results disclosed brain modular states that differentiate cognitively intact patients, patients with moderate cognitive deficits, and patients with severe cognitive deficits. Our new approach complements classical methods, offering a new way to track the brain modular states, in healthy subjects and patients, on an adequate task-specific timescale.

Assessing the accuracy of detected breaks in Landsat time series as predictors of small scale deforestation in tropical dry forests of Mexico and Costa Rica

Tracking the occurrence of deforestation events is an essential task in tropical dry forest (TDF) conservation efforts. Ideally, deforestation monitoring systems would identify a TDF clearing with near real time precision and high spatial detail, and alert park managers and environmental practitioners of illegal forest clearings occurring anywhere in a region of interest. Over the past several years there have been significant advances in the design and application of continuous land cover change mapping algorithms with these capabilities, but no studies have implemented such methods over human dominated TDF environments where small–scale deforestation (<5 ha) is widespread and hard to detect with moderate resolution sensors. The general objective for this research was to evaluate the overall accuracy of the BFASTSpatial R Package for detecting and monitoring small-scale deforestation in four sites located in tropical dry forest landscapes of Mexico and Costa Rica using greenness and moisture spectral indices derived from Landsat time series. Results show a high degree of spatial agreement (90%–94%) between the distribution of TDF clearings occurred during the 2013–2016 period (as indicated by VHR imagery interpretation) and BFASTSpatial outputs. NDMI and NBR2 had the best performance than other indices and this is evidenced by the combined overall, user's and producer's accuracies. In particular, NBR2 were the most accurate predictor of deforestation with an overall accuracy of 94.5%. Our results also imply that monitoring sites at an annual basis is feasible using BFASTSpatial and LTS, but that lower confidence should be given to sub-annual products given significant systematic temporal differences between the BFASTSpatial monthly product and reference data. The possibility of including more clear observations at the spatial resolution of Landsat (30-m) or higher will greatly increase the spatial and temporal accuracies of the method. Given its performance, BFASTSpatial can help monitor hotspots of small-scale TDF loss across Central and North America at little or no cost. Users of the method should have a strong knowledge of the local land use and land cover dynamics and the ecophysiology of vegetation types present in the landscape. This local expertise is necessary for interpreting and validating results as well as communicating its output to decision-makers and stakeholders.

Perception of time in microgravity and hypergravity during parabolic flight.

We explored the effect of gravity on the accuracy for estimating durations of 3.5, 7, and 14 s. Experiments were performed on board an Airbus A310 during parabolic flights eliciting repeated exposures to short periods of 0, 1, and 1.8 g. Two methods for obtaining duration estimates were used, reproduction and production of duration, in two conditions: a control counting condition and a concurrent reading condition. Simple reaction times were also measured to assess attention. The results showed that the temporal accuracies during the reproduction task in the concurrent reading condition were significantly underestimated in 0 g compared with 1 g. Reaction times were also longer in 0 g. However, there was no difference in duration estimates in the production tasks. These results suggest that the temporal underestimation in 0 g is caused by decreased selective attention and impaired retrieval of information in episodic memory.

Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation

Comparing with the well-known classic Cahn–Hilliard equation, the nonlocal Cahn–Hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials. On the other hand, it evidently brings more computational costs in numerical simulations, thus efficient and accurate time integration schemes are highly desired. In this paper, we propose two energy-stable linear semi-implicit methods with first and second order temporal accuracies respectively for solving the nonlocal Cahn–Hilliard equation. The temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly, while the spatial discretization is carried out by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are rigorously established for both methods in the fully discrete sense. Numerical experiments are conducted for a typical case involving Gaussian kernels. We test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one. In addition, long-time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay.