This paper presents a theoretical analysis of a fourth-order exponential time differencing Runge–Kutta rational approximation scheme for the Allen–Cahn equation based on real and distinct poles, aiming to establish a theoretical framework for the method combining exponential time differencing with rational approximation. Starting from the scalar linearized equation, we rigorously prove that the scheme is L-stable and that its matrix norm is stable with three types of boundary conditions, revealing that the L-acceptable property of the RDP rational approximation is the key to ensuring stability. Regarding the error analysis, we establish that the fully discrete scheme attains fourth-order temporal convergence, which theoretically confirms that the scheme achieves fourth-order accuracy. Numerical examples in both 2D and 3D verify the accuracy and effectiveness of the method.

This paper explores the formation of round and square crystals on curved dynamic substrates, employing the evolving surface phase field model as the mathematical framework and utilizing the evolving surface finite element method as the computational approach. The investigation focuses on the physical process of crystal growth on uneven, dynamic ultra-thin materials and biology thin films. The curved dynamic substrate is represented by the evolving surface. On evolving surfaces, two models, the classical phase field crystal model and the square phase field crystal model, are established and their respective physical properties are examined. Additionally, an efficient numerical simulation technique is devised by integrating the evolving surface finite element method with a time semi-implicit scheme. To validate the models, a series of numerical simulations are conducted to examine the physical properties and dynamical behaviors of the crystals formed under various conditions.

The growing complexity of computer viruses has necessitated the development of more sophisticated mathematical models to understand and control cyber-epidemics. Building on classical compartmental approaches, we extend the Susceptible-Infected-Removed-Antidotal (SIRA) model into a reactiondiffusion framework to capture both local infection dynamics and spatial transmission across networks. The resulting system presents significant analytical challenges due to stiffness, positivity constraints, and nonlinear couplings. To address these issues, we propose a semi-implicit numerical scheme that treats diffusion terms implicitly for stability and reaction terms explicitly for computational efficiency, coupled with Nonstandard Finite Difference (NSFD) techniques to ensure solution positivity. Rigorous stability analysis based on M-matrix theory establishes conditions for boundedness and convergence of the discrete system. Compared to existing methodologies, our approach enables accurate, efficient simulation of large-scale malware propagation, offering critical insights for cybersecurity strategies such as antivirus deployment optimization and outbreak containment. Numerical experiments demonstrate the effectiveness of the proposed scheme in capturing complex spatiotemporal dynamics of cyber threats.

In this paper, an immersed finite element method is proposed for solving parabolic problems on surfaces with static interfaces. Immersed finite element basis functions are constructed based on the triangular surface finite element. The advantage of the proposed method is that it allows interface across triangular elements, thereby avoiding the generation of complex fitted meshes on surfaces. Error estimates are derived for both the semi-discrete and fully-discrete schemes of the proposed method. Finally, some numerical examples are presented to illustrate the accuracy of the proposed method.

A local meshless radial basis function-finite difference (RBF-FD) method, leveraging polyharmonic spline (PHSn) kernel, is introduced for solving elliptic interface problems with non-homogeneous jump conditions on curved surfaces. The design of the proposed method is based on the splitting of surface region according to the interface, and the quasi-uniformed meshless fitted nodes on the interface and surface. The meshless fitted setting ensures the efficiency of scattered node generation and the simplicity of discretization. Additionally, we ensure the positivity of the basis functions through polyharmonic spline-coupled polynomials, thus making the interpolation matrix invertible. By incorporating the shape parameter-free PHSn function within the RBF-FD framework, our method yields remarkable spatial accuracy, matrix sparsity, and high-order polynomial convergence of second-, third-, and fourth-order in discretizing the surface interface problem. In the numerical experiments section, we comprehensively demonstrate the superiority of our method in terms of accuracy through a series of rigorous tests. Additionally, we explore the influence of diverse factors, including stencil size, polynomial degree, polyharmonic spline exponent, quasi-uniformed scattering distance, and the diffusion coefficient, on the overall efficiency of our method.

The Monge-Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.

In this paper, we propose a new second-order variable stepsize time-filtering algorithm for simulating natural convection phenomena. Unlike existing methods, which are primarily confined to constant stepsize frameworks, the proposed algorithm introduces a variable stepsize scheme characterized by simple implementation, high computational efficiency, and superior convergence accuracy. Furthermore, to address the practical requirements of low-memory solvers, the algorithm is extended to an adaptive variable stepsize and variable order framework for fluid flow simulations. The key theoretical contribution lies in the rigorous proof of the algorithm’s unconditional stability and second-order convergence under variable stepsize conditions. This analysis resolves a critical limitation of conventional constant stepsize methods–their inherent inflexibility in handling multiscale temporal dynamics and complex flow regimes. By leveraging adaptive stepsize control, our method not only overcomes the theoretical constraints of constant stepsize approaches but also ensures robust performance in scenarios involving sharp gradients or transient behaviors. Finally, the effectiveness and computational efficiency of the algorithm are validated through numerical experiments.

This work is devoted to the Oseen iterative stabilized difference finite element (SDFE) method for the three-dimensional (3D) steady incompressible magnetohydrodynamical (MHD) equations. Our interest in the numerical analysis of the SDFE method stems from the following two aspects: First, difference finite element method overcomes the difficulty of the 3D space discretization. The essence of this method resides in adopting the finite difference discretization in the z-direction and the finite element discretization in the (x,y) plane. This approach can maximize the good applicability and efficiency of both finite difference method and finite element method. Second, the stabilized method we use for approximating the problem in (x,y) plane does not require the specification of mesh-dependent parameters and retains the symmetry of the original equations. The rigorous analysis of uniform stability and uniform convergence is conducted, demonstrating that the SDFE method is stable and has optimal convergence. Several numerical tests are provided, confirming the theoretical predictions and verifying the accuracy of the considered method.

This study proposes an efficient numerical scheme for solving nonlinear reaction-diffusion equations. The scheme employs a fourth-order compact finite difference scheme for spatial discretization and couples it with a fourth-order exponential time differencing Runge-Kutta method to achieve a fully discrete formulation. Furthermore, dimensional splitting is combined with the Pad & eacute;(2,2) rational approximation to enhance the computational efficiency of exponential matrix evaluations. A rigorous stability analysis of the scheme is provided, along with a derivation of the optimal error estimate of order O(k4). Numerical experiments demonstrate that the proposed scheme maintains accuracy even with coarser grid configurations.

This paper addresses the challenge of weak initial singularities in solutions to the nonlinear time-fractional Schr & ouml;dinger (NTFS) equation. We construct and analyze a novel numerical scheme based on a coupled time-space discretization strategy, which combines the L1 scheme on graded meshes with the Virtual Element Method (VEM). Temporally, the L1 scheme on graded meshes is adopted to effectively approximate the Caputo derivative, while nonlinearity is efficiently handled via the scalar auxiliary variable (SAV) approach combined with the Crank-Nicolson (CN) format. This leads to the establishment of a novel fully discrete numerical scheme termed L1-CN-VEM. We rigorously prove the convergence of the proposed scheme and derive corresponding error estimates, thereby providing a solid theoretical foundation for solving this class of singular and complex problems.

Roll coating (RC) is a fundamental process in both industrial and decorative applications, including the production of wallpapers, plastic and photographic films, adhesive tapes, magnetic recording media, and packaging materials. In this study, we develop a mathematical model for the flow of Powell-Eyring fluid confined within a narrow gap formed between a moving roll and a fixed substrate. An analytical framework based on the perturbation method is constructed to obtain approximate solutions for the velocity field, temperature distribution, pressure gradient, and pressure profile. A comprehensive comparison between numerical and analytical results is presented, validating the accuracy and reliability of the proposed formulations. Parametric analysis highlights the impact of key material and flow properties on coating thickness, velocity, pressure distribution, temperature variation, separation force, power input, and Nusselt number. The maximum coating thickness attains at , while the minimum value is at . Nusselt number increases with higher Brinkman number, signifying stronger heat transfer effects. A mechanism for controlling the coating thickness, power input, separation force, Nusselt number, and pressure distribution is provided by the material properties involved, offering practical insights for optimizing RC operations across diverse industrial applications.

A computer virus poses significant risks to individual computer systems. To mitigate these risks, various mathematical models have been developed. Several techniques, including the installation of antivirus software and the implementation of preventive measures based on epidemiological studies, can reduce the impact of virus attacks. This study focuses on the reaction-diffusion computer virus model. A qualitative analysis of the model is conducted, and the positivity of the model is established. Additionally, the boundedness of the model’s solutions is demonstrated. The stability region is determined by analyzing the variational matrix across different parametric spaces for both temporal and diffusive components of the model. It is observed that the stability region expands under the influence of diffusion phenomena. For the numerical investigation of the model, three computational schemes are employed: the forward Euler scheme, the backward Euler operator splitting scheme, and the non-standard finite difference (NSFD) scheme. The NSFD scheme is analyzed in terms of positivity, consistency, and convergence, with positivity being proven using M-matrix theory. A test problem is utilized to obtain numerical solutions, and various parameter values are explored to identify virus-free and virus-endemic equilibrium states. The NSFD scheme is shown to preserve all essential characteristics of the continuous model.

In this paper, we study unconditional structure preserving fully discrete finite element method (FEM) for the Keller-Segel equations on closed surfaces. Based on the equivalent form got from the Slotboom transformation, applying the surface finite element method (SFEM) and the Euler scheme in space and time, we present a fully discrete scheme for the Keller-Segel equations. Then, by modifying the approximation of the exponential function with piecewise local geometric means and introducing two auxiliary variables with respect to the Slotboom transformation to split the coupling, and applying the lump process to the mass matrix, we get a new structure preserving fully discrete finite element scheme. The proposed scheme is rigorously proved to be stable, unconditionally mass conservative, positivity preserving and energy dissipative. Moreover, at each time step, only the linear elliptic equation with constant coefficient needs to be solved, which can be implemented efficiently. Finally, a number of numerical experiments are shown to demonstrate the theoretical analysis.

The Cahn-Hilliard(C-H) equation is a key phase-field model for describing nonlinear interface dynamics in multiphase flows. During its evolution, the equation demonstrates significant energy variation features: the energy change tends to stabilize as the system approaches a steady state, while it exhibits pronounced fluctuations during phase transitions or coarsening stages. Traditional numerical methods often struggle to balance computational accuracy and efficiency when solving this equation, while also maintaining its physical properties such as energy dissipation and mass conservation. To tackle these challenges, this research introduces a surrogate model based on the Long Short-Term Memory (LSTM) neural network to enhance computational efficiency. Specifically, we utilize a large amount of time-series data with known solutions to train the LSTM network, which is then employed for short-term predictions during the energy smoothing phase. By integrating traditional numerical methods with the improved LSTM network, alternating computations are performed based on energy variations, reducing the iterative computational load and achieving the simulation acceleration of the C-H equation. Furthermore, to ensure that the evolution process adheres to the physical properties of energy dissipation and mass conservation, this study introduces a custom activation function with mass-conserving properties and a loss function that guarantees energy dissipation. These enhancements effectively improve the numerical stability and physical consistency of the model. Experimental results demonstrate that the proposed surrogate method enhances computational efficiency to a certain degree while strictly maintaining physical properties such as energy dissipation and mass conservation, providing a novel solution for complex phase-field dynamics problems.

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising tool for solving combustion problems. However, accurately modeling combustion remains challenging due to the inherent strong nonlinearities and multiscale phenomena characteristic of flame dynamics. This paper introduces an enhanced PINN framework integrated with an attention mechanism and a Radial Basis Function (RBF)-based feature mapping module. Within this architecture, the RBF module provides a set of overlapping localized feature mappings across the spatial domain, which are further processed by a self-attention-based residual module to model the complex nonlinear coupling between physical variables. When combined with adaptive activation functions and a refined feedforward network, and with the incorporation of a small set of data to ensure reliable convergence under strong nonlinearities, this approach improves the network’s ability to represent fine-scale variations and critical physical structures. Validation across one-dimensional and two-dimensional flame benchmarks demonstrates that the proposed method accurately predicts key combustion variables and reconstructs complex field distributions. Furthermore, the framework enhances the stability and convergence of multiscale physical residual optimization. Overall, this work provides a robust and flexible PINN-based methodology for high-fidelity combustion modeling.

High-order discontinuous Galerkin spectral element method (DGSEM) often encounters shock instability or carbuncle phenomenon when handling compressible flow problems with discontinuities. To solve this problem, a novel subcell limiting strategy satisfying the discrete entropy inequality is proposed. Firstly, an entropy-stable shock-capturing scheme is developed based on sign-preserving interpolation of entropy variables and an entropy stable flux. Then, the shock capturing scheme is used in subcell limiting for high-order entropy-stable DGSEM based on Legendre-Gauss solution points to prevent the carbuncle phenomenon. Both discontinuous and continuous blending functions are applied to combine the two schemes. The entropy-stable DGSEM with the subcell entropy-stable shock capturing scheme is, in fact, a hybrid scheme. The discrete conservation laws and discrete entropy inequality are proved theoretically for the proposed hybrid scheme. Numerical investigations show that the proposed schemes satisfy both discrete conservation laws and discrete entropy inequality, achieve high resolution in smooth regions, have strong shock-capturing capability, and can effectively prevent the carbuncle phenomenon.

The phase-field simulation is quite sensitive to the settings of model parameters and initial conditions, as different settings may yield significantly different simulation results. However, in practical phase-field simulation, model errors may arise from the initial input to the model. Therefore, utilizing available real-time observations for data assimilation to enhance the accuracy of numerical simulation has become an important research topic. Traditional data assimilation methods, such as three-dimensional variational data assimilation and Ensemble Kalman Filter, face challenges due to the need for computing high-dimensional covariance matrices or solving high-dimensional optimization problems, which result in low computational efficiency and high storage requirements. To address these challenges, this paper proposes and compares three parallel data assimilation methods based on the operator splitting method: nudging, three-dimensional variational data assimilation, and Ensemble Kalman Filter. Although the three methods all reduce computational costs and storage requirements, each has its own specific advantages. By comparing the three methods and integrating their strengths, this paper further proposes a more comprehensive hybrid data assimilation method, which significantly improves simulation accuracy and avoids the limitations of using a single data assimilation method. Meanwhile, the effectiveness of the hybrid method in improving simulation accuracy is validated using the phase-field dendritic growth model as a test case.

A one-stage high-order and high-resolution hybrid scheme is proposed in this paper to solve the Euler equations for compressible shocked flows. It is based on the framework of the one-stage efficient high-order gas kinetic scheme (EHGKS), which achieves consistent fifth order of accuracy in both space and time, and newly-developed function-wise compact reconstruction technique, hybridizing the simple Hermite weighted essentially non-oscillatory (SHWENO) reconstruction technique with the self-adjusting steepness-based (SAS) limiter. The original fifth-order polynomial of SHWENO is adopted in the new reconstruction technique preserve the high-order accuracy and works in the smooth regions; while the low-order polynomials to preserve robustness in the original SHWENO have been replaced by a new one, which is extended from the SAS limiter, and they only work near the discontinuities. The original smoothness indicators of SHWENO and SAS are combined to detect the shock and the weight of each high-order and high-resolution counterpart. The final polynomial returns to the SAS limiter and self-adjusts its steepness parameter near the discontinuities. The new reconstruction technique enables the original compact and one-stage high-order scheme EHGKS-SHWENO further with a sharp resolution for the discontinuities. A series of numerical tests are carried out to validate the accuracy and resolution of the improved scheme. The numerical results demonstrate that the new scheme preserves the desired fifth order of accuracy, while maintaining a sharper resolution for the discontinuities than EHGKS-SHWENO.

The Q1-finite element spaces, in any finite d-dimension, are equipped with the discrete L'2h-inner product generated by the simple row-sum mass lumping. The equivalence of the discrete L'2h-norm and the L2-norm on the Q1-finite element spaces is uniform in mesh size h, in both cases of uniform and nonuniform partitions. Several representation formulae for these norms are derived. Using these, accurate bounds between these two norms are obtained, which is our major contribution. Examples show that these bounds are sharp. As an important application, the equivalence is established between discrete h1h-norm and H1-norm. Numerical results are presented.

We propose an hp-adaptive Galerkin least squares intrinsic finite element method for the stationary convection-reaction-diffusion equation on closed surfaces. The discrete variational formulation is intrinsic to the interpolated surface, incorporating the contribution of the local parametrization. This eliminates the reliance on pre-established surface parametrization and circumvents the challenge of high-order numerical integration in the space where the surface is embedded. The Galerkin least squares method, a variant of the streamline-upwind/Petrov-Galerkin method, is employed to enhance the stability of the numerical scheme. Unique solvability of the discrete variational problem is shown. To generate appropriate anisotropic mesh, an hp-adaptive refinement strategy is designed by combining the Galerkin least squares method with the quadratic intrinsic finite element method. The recovery function, based on centroid weights, yields more accurate estimates compared to area-weighted counterparts. Finally, numerical experiments show the effectiveness of the proposed method.