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Abstract Enhancing crop water productivity is crucial for regional water resource management and agricultural sustainability, particularly in arid regions. However, evaluating the spatial heterogeneity and temporal dynamics of crop water productivity in face of data limitations poses a challenge. In this study, we propose a framework that integrates remote sensing data, time series generative adversarial network (TimeGAN), dynamic Bayesian network (DBN), and optimization model to assess crop water productivity and optimize crop planting structure under limited water resources allocation in the Qira oasis. The results demonstrate that the combination of TimeGAN and DBN better improves the accuracy of the model for the dynamic prediction, particularly for short-term predictions with 4 years as the optimal timescale (R2 > 0.8). Based on the spatial distribution of crop suitability analysis, wheat and corn are most suitable for cultivation in the central and eastern parts of Qira oasis while cotton is unsuitable for planting in the western region. The walnuts and Chinese dates are mainly unsuitable in the southeastern part of the oasis. Maximizing crop water productivity while ensuring food security has led to increased acreage for cotton, Chinese dates and walnuts. Under the combined action of the five optimization objectives, the average increase of crop water productivity is 14.97%, and the average increase of ecological benefit is 3.61%, which is much higher than the growth rate of irrigation water consumption of cultivated land. It will produce a planting structure that relatively reduced irrigation water requirement of cultivated land and improved crop water productivity. This proposed framework can serve as an effective reference tool for decision-makers when determining future cropping plans.

Background: The roll-coating process plays an important role in many industries for its practical applications, such as paint, PVC-coated fabrics and plastic industries. Roll coating is commonly used to put thin coating films over continuous substrates such as foils and papers. There are several roll-to-roll coating methods, including forward and roll over web. However, the reverse roll coating study of the non-Newtonian fluid model is presented in this paper. The constitutive equation for the viscoelastic Carreau fluid model as the non-Newtonian fluid is derived using the momentum and energy equations. To simplify the nonlinear systems of partial differential equations, appropriate non-dimensional parameters are employed to convert them into systems of ordinary differential equations. Method: By taking the Weissenberg number as a small parameter, a series solution is obtained for considerable quantities like pressure gradient, axial velocity, flow rate, pressure distribution, and temperature distribution using the perturbation method. In contrast, the numeric outcome of some quantities of engineering interest, like flow rate, coating thickness, share stresses, and separation points, are calculated. Streamlines in 2D and 3D are also drawn to observe the flow pattern. To validate the solution, the problem is also solved numerically with the BVP Midrich method. The analysis of the results demonstrates a satisfactory agreement between the analytical and numerical solutions. Results: The graphical investigation examines the influence of different fluid parameters on velocity profile, shear stress, pressure profile, temperature profile, and pressure gradient. A comprehensive mechanism behind these outcomes is deliberated. It has been observed that velocity rises on increasing the Weissenberg number; however, velocity decreases in the case of the velocities ratio. Further, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Moreover, the separation point and coating thickness decrease with the velocities ratio increase. The results accuracy is checked against papers that have already been published. The comparison reveals good agreement; therefore, the resultant analytical model gives new ways to forecast film thickness and explain influence parameters. Hence, these factors may help in an efficient coating process and improve the substrate life.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

Multi-object pedestrian tracking plays a crucial role in autonomous driving systems, enabling accurate perception of the surrounding environment. In this paper, we propose a comprehensive approach for pedestrian tracking, combining the improved YOLOv8 object detection algorithm with the OC-SORT tracking algorithm. First, we train the improved YOLOv8 model on the Crowdhuman dataset for accurate pedestrian detection. The integration of advanced techniques such as softNMS, GhostConv, and C3Ghost Modules results in a remarkable precision increase of 3.38% and an mAP@0.5:0.95 increase of 3.07%. Furthermore, we achieve a significant reduction of 39.98% in parameters, leading to a 37.1% reduction in model size. These improvements contribute to more efficient and lightweight pedestrian detection. Next, we apply our enhanced YOLOv8 model for pedestrian tracking on the MOT17 and MOT20 datasets. On the MOT17 dataset, we achieve outstanding results with the highest HOTA score reaching 49.92% and the highest MOTA score reaching 56.55%. Similarly, on the MOT20 dataset, our approach demonstrates exceptional performance, achieving a peak HOTA score of 48.326% and a peak MOTA score of 61.077%. These results validate the effectiveness of our approach in challenging real-world tracking scenarios.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre–Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.

Abstract In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O ( τ 2 − α + h 4 ) $O(\tau ^{2-\alpha }+h^{4})$ and O ( τ 3 − α + h 4 ) $O(\tau ^{3-\alpha }+h^{4})$ , respectively, with the temporal step size τ and the spatial step size h, and α ∈ ( 0 , 1 ) $\alpha \in (0,1)$ . For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier–Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier–Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration.

With the remarkable development of deep learning in the field of science, deep neural networks provide a new way to solve the Stefan problem. In this paper, deep neural networks combined with small sample learning and a general deep learning framework are proposed to solve the two-dimensional Stefan problem. In the case of adding less sample data, the model can be modified and the prediction accuracy can be improved. In addition, by solving the forward and inverse problems of the two-dimensional single-phase Stefan problem, it is verified that the improved method can accurately predict the solutions of the partial differential equations of the moving boundary and the dynamic interface.

Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has successfully solved forward and inverse PDE problems. In this study, we considered a parametric light wave equation, discretized it using the central difference, and, through this difference scheme, constructed a new neural network structure named the second-order neural network structure. Additionally, we used the adaptive activation function strategy and gradient-enhanced strategy to improve the performance of the neural network and used the deep mixed residual method (MIM) to reduce the high computational cost caused by the enhanced gradient. At the end of this paper, we give some numerical examples of nonlinear parabolic partial differential equations to verify the effectiveness of the method.

In this paper, we study the finite element method of the Navier–Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.

This paper studies the energy stability property of the correction procedure via reconstruction (CPR) method with staggered flux points based on second-order subcell limiting. The CPR method with staggered flux points uses the Gauss point as the solution point, dividing flux points based on Gauss weights, with the flux points being one more point than the solution points. For subcell limiting, a shock indicator is used to detect troubled cells where discontinuities may exist. Troubled cells are calculated by the second-order subcell compact nonuniform nonlinear weighted (CNNW2) scheme, which has the same solution points as the CPR method. The smooth cells are calculated by the CPR method. The linear energy stability of the linear CNNW2 scheme is proven theoretically. Through various numerical experiments, we demonstrate that the CNNW2 scheme and CPR method based on subcell linear CNNW2 limiting are energy-stable and that the CPR method based on subcell nonlinear CNNW2 limiting is nonlinearly stable.

Identifying suitable habitats for endangered species is critical in order to promote their recovery. However, conventional species distribution models (SDMs) need large amounts of labeled sample data to learn the relationship between species and environmental conditions, and are difficult to fully detangle the role of the environment in the distribution of the endangered species, which are very sparsely distributed and have environmental heterogeneity. This study’s first innovation used the semi-supervised model to accurately simulate the suitable habitats for endangered species with a small sample size. The model performance was compared with three conventional SDMs, namely Maxent, the generalized linear model, and a support vector machine. Applying the model to the endangered species Populus euphratica (P. euphratica) in the lower Tarim River basin (TRB), Northwest China. The results showed that the semi-supervised model exhibited better performance than conventional SDMs with an accuracy of 85% when only using 443 P. euphratica samples. All models developed using smaller sample sizes exhibit worse performance in the prediction of habitat suitability areas for endangered species while the semi-supervised model is still excellent. The results showed that the suitable habitat for P. euphratica is mainly near the river channel of the lower TRB, accounting for 13.49% of the study area. The lower Tarim River still has enormous land potential for the restoration of endangered P. euphratica . The model developed here can be used to evaluate a suitable habitat for endangered species with only a small sample size, and provide a basis for the conservation of endangered species.

In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.

When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, which can balance the gradients of each loss term during training. The weight is updated by the idea of the minmax algorithm. The neural network identifies which types of training data are harder to train and forces it to focus on those data before training the next step. Specifically, it adjusts the weight of the data that are difficult to train to maximize the objective function. On this basis, one can adjust the network parameters to minimize the objective function and do this alternately until the objective function converges. We demonstrate that the dynamic weights are monotonically non-decreasing and convergent during training. This method can not only accelerate the convergence of the loss, but also reduce the generalization error, and the computational efficiency outperformed other state-of-the-art PINNs algorithms. The validity of the method is verified by solving the forward and inverse problems of the Navier–Stokes equation.

In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify the accuracy and effectiveness of the proposed method.

In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “∇·u=0”, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter ϵ, the time-step size τ, and the mesh size h. Finally, two numerical tests show that our scheme is effective.

We propose and analyze an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations in this paper. The proposed method is a first-order velocity correction projection algorithms in time marching, including standard velocity correction and rotation velocity correction, which can completely decouple all variables in the model. Meanwhile, the schemes are not only linear and only need to solve a series of linear partial differential equations with constant coefficients at each time step, but also the standard velocity correction algorithm can produce the Neumann boundary condition for the pressure field, but the rotational velocity correction algorithm can produce the consistent boundary which improve the accuracy of the pressure field. Thus, improving our computational efficiency. Then, we give the energy stability of the algorithms and give a detailed proofs. The key idea to establish the stability results of the rotation velocity correction algorithm is to transform the rotation term into a telescopic symmetric form by means of the Gauge–Uzawa formula. Finally, numerical experiments show that the rotation velocity correction projection algorithm is efficient to solve the thermally coupled magneto-hydrodynamic equations.

In this paper, a grid-free deep learning method based on a physics-informed neural network is proposed for solving coupled Stokes–Darcy equations with Bever–Joseph–Saffman interface conditions. This method has the advantage of avoiding grid generation and can greatly reduce the amount of computation when solving complex problems. Although original physical neural network algorithms have been used to solve many differential equations, we find that the direct use of physical neural networks to solve coupled Stokes–Darcy equations does not provide accurate solutions in some cases, such as rigid terms due to small parameters and interface discontinuity problems. In order to improve the approximation ability of a physics-informed neural network, we propose a loss-function-weighted function strategy, a parallel network structure strategy, and a local adaptive activation function strategy. In addition, the physical information neural network with an added strategy provides inspiration for solving other more complicated problems of multi-physical field coupling. Finally, the effectiveness of the proposed strategy is verified by numerical experiments.

In this paper, we present a fully Lagrangian method based on the radial basis function (RBF) finite difference (FD) method for solving convection–diffusion partial differential equations (PDEs) on evolving surfaces. Surface differential operators are discretized by the tangent plane approach using Gaussian RBFs augmented with two-dimensional (2D) polynomials. The main advantage of our method is the simplicity of calculating differentiation weights. Additionally, we couple the method with anisotropic RBFs (ARBFs) to obtain more accurate numerical solutions for the anisotropic growth of surfaces. In the ARBF interpolation, the Euclidean distance is replaced with a suitable metric that matches the anisotropic surface geometry. Therefore, it will lead to a good result on the aspects of stability and accuracy of the RBF-FD method for this type of problem. The performance of this method is shown for various convection–diffusion equations on evolving surfaces, which include the anisotropic growth of surfaces and growth coupled with the solutions of PDEs.

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.

Abstract In this paper, firstly, we solve the linear 3D Schrödinger equation using Douglas–Gunn alternating direction implicit (ADI) scheme and high-order compact (HOC) ADI scheme, which have the order O(τ2+h2) $O(\tau^{2}+h^{2})$ and O(τ2+h4) $O(\tau^{2}+h^{4})$, respectively. Secondly, a fourth-order compact ADI scheme, based on the Douglas–Gunn ADI scheme combined with second-order Strang splitting technique, is proposed for solving 3D nonlinear Schrödinger equation. Stability analysis has demonstrated that these schemes are unconditionally stable. Finally, numerical results show that these schemes preserve the conservation laws and provide accurate and stable solutions for the 3D linear and nonlinear Schrödinger equations.

In this paper, we propose an adaptive defect-correction method for natural convection (NC) equations. A defect-correction method (DCM) is proposed for solving NC equations to overcome the convection dominance problem caused by a high Rayleigh number. To solve the large amount of computation and the discontinuity of the gradient of the numerical solution, we combine a new recovery-type posteriori estimator in view of the gradient recovery and superconvergent theory. The presented reliability and efficiency analysis shows that the true error can be effectively bounded by the recovery-based error estimator. Finally, the stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations.

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

In this article, we present two-grid stable mixed finite element method for the 2D Burgers’ equation approximated by the -P1 pair which satisfies the inf–sup condition. This method consists in dealing with the nonlinear system on a coarse mesh with width H and the linear system on a fine mesh with width h << H by using Crank–Nicolson time-discretization scheme. Our results show that if we choose H2 = h this method can achieve asymptotically optimal approximation. Error estimates are derived in detail. Finally, numerical experiments show the efficiency of our proposed method and justify the theoretical results.

We discuss several stabilized finite element methods, which are penalty, regular, multiscale enrichment, and local Gauss integration method, for the steady incompressible flow problem with damping based on the lowest equal-order finite element space pair. Then we give the numerical comparisons between them in three numerical examples which show that the local Gauss integration method has good stability, efficiency, and accuracy properties and it is better than the others for the steady incompressible flow problem with damping on the whole. However, to our surprise, the regular method spends less CPU-time and has better accuracy properties by using Crout solver.