A high-order compact modified Crank-Nicolson scheme is proposed for the nonlinear Sobolev equation, in which a convex energy splitting approach is employed to address key computational challenges. By decomposing the energy functional into two convex components, a spatially fourth-order and temporally second-order compact finite difference discretization that naturally preserves the physical energy structure is developed. Rigorous theoretical analysis establishes the boundedness and unique solvability of the numerical solution and proves a discrete energy dissipation law. Furthermore, the scheme is proven to be unconditionally convergent (without any time-space mesh-ratio restriction) in the discrete norm. Numerical experiments confirm that the expected convergence rates are achieved, and the scheme displays long-term stability and consistent energy decay. These results suggest that the proposed method provides an accurate, efficient, and structure-preserving numerical framework for complex nonlinear Sobolev-type problems.
This paper investigates the global $L^2$ superconvergence of a family of quadratic finite volume schemes on tetrahedral meshes. By constraining the dual mesh with orthogonal conditions, we consider a family of quadratic finite volume schemes. We note that the orthogonal conditions are also fundamental for achieving optimal $L^2$ error estimates. First, we prove the weak estimate of the second type, which plays an important role in deriving the global $L^2$ superconvergence. We then prove the global superconvergence result $||u_h−I^2_h u||_0 = \mathcal{O}(h^4),$ where $u_h$ is the numerical solution of the quadratic finite volume scheme and $I^2_hu$ is the Lagrange quadratic finite element interpolant of the exact solution. Numerical experiments verify our theoretical results and also demonstrate that finite volume schemes formulated without the orthogonal conditions on the dual mesh fail to exhibit superconvergence and, in fact, achieve only a convergence order $\mathcal{O}(h^2)$ consistent with the $H^1$ error.
Airfoils are frequently designed using empirical knowledge and practical aeromodelling experience. In many cases, essential airfoil data exists only as discrete coordinate points obtained from scanned drawings or digitized outlines. Without a parametric representation, such airfoils are challenging to analyze, optimize, or reproduce during computational fluid dynamics simulations. Motivated by the need for a universal parameterization method that can express manually designed airfoils through a consistent set of mathematical functions or parameters, a new airfoil parameterization technique based on available coordinate data is proposed in this work. The proposed method is based on the least squares error approach and is applicable to any airfoil type, making it simple, versatile, and practical. The methodology is demonstrated on a variety of airfoils including natural laminar flow (NLF) airfoil, where the reconstructed geometry is compared against the original shape. Flow simulations were then carried out on the reconstructed NLF airfoils using an in-house compressible flow solver, and the results were validated with experimental data. A close agreement between both sets of results confirms the accuracy and effectiveness of the proposed method. Additionally, the study examines how variations in airfoil curvature influence the pressure gradient variation on the airfoil surfaces and corresponding effects on the flow separation as a function of the angle of attack.
Subspace clustering is a fundamental problem in machine learning that has attracted considerable attention in recent years. Most existing methods focus on designing effective models to regularize the coefficient matrix, often neglecting the impact of noise on subspace structures. However, real-world data are typically corrupted by noise, which can distort the underlying subspace structure. Additionally, for sequential data, a key challenge is the effective exploitation of temporal information. To address these issues, we propose a novel and robust sequential subspace clustering method, termed joint capped $ℓ_2$ and $ℓ_{2,p}$ norm minimization (JCLLM). The capped $ℓ_2$ norm-based loss function mitigates the influence of noise and outliers in regression, while the $ℓ_{2,p}$ norm regularization captures the temporal dependencies inherent in sequential data. We develop an iteratively reweighted optimization algorithm to solve the JCLLM model and prove its convergence to a stationary point. Extensive experiments on both synthetic and real-world datasets demonstrate that our method consistently outperforms several state-of-the-art subspace clustering approaches.
The simulation of nano scale semiconductor devices relies on the coupled Poisson-Schro & uml;dinger equations that capture the interaction between surface potential and wave function. Over the past two decades many researchers have proposed various analytical and computational approaches to analyze the solution of the Poisson-Schro & uml;dinger system and to deepen their understanding. In this study, we present an iterative procedure utilizing a finite difference - cubic spline approach that gives self consistent solution to these equations. We performed a numerical analysis of the finite difference - cubic spline based iterative approach to estimate electron concentration calculated from the solution of the Schro & uml;dinger equation as a wave function. We followed a singular perturbation approach to solve the above system as one of the equations is of reaction-diffusion type. Also, detailed error estimation is carried out, resulting in a higher order convergence. Numerical experiments with Neumann boundary conditions are included to validate our theoretical results.
Inspired by widespread attention to the frequency of individual daily travel and the number of visitors at the place where individuals meet in infectious disease prevention and control work, we establish a class of the SIS epidemic model involving these two factors in a way that differs meaningfully from the traditional model. We first calculate the basic reproduction number of the model. Next, we analyse the global dynamics of the model by applying monotone dynamical systems theory. Then, we show that the increase in travel frequency of individuals will increase their risk of being infected. Both the heterogeneity in the individual’s moving pattern and the visitor distribution in the different venues play important roles in the propagation process. By comparing different types of control measures we show that their effectiveness depends on the distribution of visitor volume.
In this paper, we propose and develop a novel variational model based on hue-saturation similarity and fuzzy membership function for color image segmentation. The main contribution of the proposed model is that we determine different segments by using the similarity of hue and saturation information in hue, saturation, and value color space. We first provide specific definitions of the hue/saturation distance to describe hue-saturation similarity, then formulate a novel data fitting term with an adaptive weight coefficient by using hue-saturation similarity in the proposed energy functional. Two efficient iterative algorithms based on coordinate descent method and alternating direction method of multipliers have been proposed to solve the proposed optimization problem. Theoretically we study the existence of the solution of the proposed model and the convergence of the proposed coordinate descent algorithm. Numerical experimental results demonstrate that the segmentation performance of the proposed model is much better than that of other existing color image segmentation methods.
Subspace clustering is a fundamental problem in machine learning that has attracted considerable attention in recent years. Most existing methods focus on designing effective models to regularize the coefficient matrix, often neglecting the impact of noise on subspace structures. However, real-world data are typically corrupted by noise, which can distort the underlying subspace structure. Additionally, for sequential data, a key challenge is the effective exploitation of temporal information. To address these issues, we propose a novel and robust sequential subspace clustering method, termed joint capped 2 pound and 2 pound,p norm minimization (JCLLM). The capped 2 pound norm-based loss function mitigates the influence of noise and outliers in regression, while the 2 pound,p norm regularization captures the temporal dependencies inherent in sequential data. We develop an iteratively reweighted optimization algorithm to solve the JCLLM model and prove its convergence to a stationary point. Extensive experiments on both synthetic and real-world datasets demonstrate that our method consistently outperforms several state-of-the-art subspace clustering approaches.
The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.
We investigate the stability of the ternary Allen-Cahn (tAC) equation solved by the fully explicit Euler method. The tAC equation models interface dynamics and phase separation in three-component systems, and is formulated as an L2-gradient flow of the Ginzburg-Landau energy. We derive a sufficient condition on the time step size that satisfies the discrete maximum principle and guarantees the numerical stability of the fully explicit Euler method. Through detailed analysis, we confirm that the derived condition provides the largest possible time step size among those that preserve stability. Numerical experiments are conducted to verify the theoretical results, and the numerical solution violates the maximum principle and becomes unstable when the time step size exceeds the derived bound. Conversely, when the time step condition is satisfied, the computational solution preserves the maximum principle and maintains the energy dissipation property of the tAC equation. These results provide a rigorous theoretical foundation for determining the stability limit of the explicit Euler method applied to the tAC equation.
In this paper, we investigate numerical methods for computing the ground state of space-fractional nonlinear Schr & ouml;dinger equation. We focus on extending the locally optimal preconditioned conjugate gradient method, originally designed for linear eigenvalue problems, to address this nonlinear fractional framework. Through comparative numerical experiments with the discrete gradient flow method, we demonstrate that the locally optimal preconditioned conjugate gradient method achieves significantly superior efficiency in many scenarios. This advantage is particularly pronounced in multidimensional problems and cases involving lower fractional derivative orders. The results highlight the adaptability of the locally optimal preconditioned conjugate gradient approach to eigenproblem of fractional nonlinear systems, offering a robust and high efficient computational alternative for highdimensional or low-order fractional derivative settings.
The stationary Wigner equation (SWE) is often used to model the quantum transport in semiconductor devices. The discrete stationary Wigner equation is derived using the first-order upwind scheme and the sinc-Galerkin method [H. Jiang, T. Lu, and W. Zhang, J. Comput. Appl. Math. 409 (2022), 114152]. Based on the successive over relaxation (SOR) iterative method, we develop the basic SOR-like (BSOR-like) and updated SOR-like (USOR-like) iterative methods for solving the discrete stationary Wigner equation efficiently. The main difference between these two iterative methods is that the USOR-like iterative method aims to make more use of the updated components of the Wigner function by splitting the pseudo-differential term. The convergence range of the relaxation parameter in the USOR-like iterative method is numerically investigated. Compared with that of the BSOR-like iterative method, this interval is significantly enlarged. Numerical results have also shown that the USOR-like iterative method is more computationally efficient than the BSOR-like iterative method. As an application, the resonant tunneling effect and the effects of scattering are investigated by simulating a resonant tunneling diode (RTD) using the USOR-like iterative method.
We present a new space-time extrapolation cascadic multigrid method with SSOR preconditioned GMRES smoother for linear parabolic equations. This method simultaneously solves all time steps of the system using an all-at-once approach, employing Crank-Nicolson discretization in time and central difference discretization in space. The key components of the new algorithm are the Richardson extrapolation and Lagrange interpolation operators. By utilizing these techniques with numerical solutions of current and previous grids, we generate a good initial guess for the iterative solution on the next finer grid, greatly reducing the number of required iterations and computational time. Finally, we explain how to implement the new multigrid method and show its efficiency through numerical experiments.
Dirichlet-Neumann and Neumann-Neumann methods are not only the parallel strategies in the spatial domain, but also can be used as a class of parallel methods in time. In this paper, we propose the Dirichlet-Neumann and Dirichlet-Neumann algorithms for time-periodic parabolic optimal control problems. By the Lagrange multiplier approach, a coupled system is obtained with a special time-periodic condition. For this coupled system, the Dirichlet-Neumann and Neumann-Dirichlet algorithms and their three variants are derived. We present the convergence analysis for all proposed algorithms. The numerical performance of the convergence factors is shown to illustrate our theoretical analysis. Based on our analysis, there is a class of algorithms with the better convergence compared with the natural Dirichlet-Neumann algorithm. Finally, numerical experiments are provided to illustrate the theoretical results.
In this paper, we propose an asymptotic preserving (AP) method for the weakly nonlinear Klein-Gordon equation (NKGE). Firstly, we apply a multi-scale expansion for the weakly NKGE and obtain the equation for the leading-order term, for which an error estimate has been provided. Secondly, by solving the equation for the leading-order term numerically, we construct an AP method for the weakly NKGE. Finally, numerical results in one spatial dimension are provided to show that: (i) The method is asymptotic preserving, i.e., the error between the leading-order term and the solution of the weakly NKGE behaves as $\mathcal{O}$($ε$) as $ε$ → 0. (ii) It is uniformly accurate since the numerical solution obtained by the method is independent of the small parameter $ε$. (iii) It can make correct predictions about the solution of the original NKGE. Moreover, extension of the method to the two-dimensional weakly NKGE are also provided.
This paper focuses on the problem of recovering ternary sparse signals with s nonzero entries of 1 and −1. We propose three novel algorithms: ternary matching pursuit (TMP), ternary generalized orthogonal matching pursuit (TGOMP), and piecewise ternary generalized orthogonal matching pursuit (PTGOMP). First, inspired by the binary matching pursuit algorithm, we introduce the TMP algorithm, which assigns values of 1 or −1 based on the most correlated residual, and provide theoretical guarantees based on the mutual coherence, denoted by $µ$ and the restricted isometry property of the measurement matrix, respectively. Second, we propose the TGOMP algorithm, by selecting multiple $(M)$ indices at each iteration in order to improve the performance of the TMP algorithm. We establish a sufficient condition $µ < 1/(2s − 1)$ that ensures the TGOMP algorithm selects $M$ correct indices and corresponding entries of $x$ in each iteration. Especially, all correct indices and entries can be selected in the first iteration. Additionally, we present a sufficient condition based on the restricted isometry property that guarantees all correct indices are selected in at most s iterations. Third, we propose the PTGOMP algorithm, which employs a piecewise selection strategy at each iteration, further improves recovery performance. Theoretical guarantees for the PTGOMP algorithm are derived based on the mutual coherence, showing its advantages in ternary sparse signal recovery. Finally, we validate the effectiveness of our algorithms through simulations and numerical experiments, demonstrating that combining appropriate matrix structures with suitable sparsity patterns can significantly improve recovery performance.
We propose an adaptive sampling method for the training of Physics Informed Neural Networks (PINNs) which allows for sampling based on an arbitrary problem-specific heuristic which may depend on the network and its gradients. In particular we focus our analysis on the Allen-Cahn equations, attempting to accurately resolve the characteristic interfacial regions using a PINN without any post-hoc resampling. In experiments, we show the effectiveness of these methods over residual-adaptive frameworks.
Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemotaxis systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions in 3 space dimensions (3D). This is an expensive computation when solutions develop large gradients at unknown locations, and so efficient algorithms to capture the main dynamical behavior are valuable. Here as a case study, we consider a parabolic-hyperbolic Keller-Segel (PHKS) system in the angiogenesis literature, and develop a mesh-free particle-based neural network algorithm that scales better to 3D than traditional mesh based solvers. From a regularized approximation of PHKS, we derive a neural stochastic interacting particle-field (NSIPF) algorithm where the bacterial density is represented as empirical measures of particles and the field variable (concentration of chemo-attractant) by a convolutional neural network trained on low cost synthetic data. As a new model, NSIPF preserves total mass and non-negativity of the density, and captures the dynamics of 3D multi-bump solutions at much faster speeds compared with classical finite difference and spline based SIPF methods.
In this paper, the method of fundamental solutions (MFS) is first developed for solving direct problems in bi-layer materials in the biomedical field of optical fluorescence. The governing system of second-order linear partial differential equations (PDEs) for the emission and excitation fluences is transformed into a single fourth-order PDE with appropriate boundary and interface matching conditions. The MFS is subsequently further developed, in conjunction with a constrained minimization regularization procedure, to solve nonlinear inverse optical fluorescence tomography problems. Numerical results confirm the accuracy, stability and versaof the meshless technique
Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, and the nonlinear Schrödinger equations, the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the definition of the discrete variational derivative, based on which, a novel semi-analytical multiple invariants-preserving integrator for the conservative partial differential equations is constructed by projection technique. The proposed integrators are shown to have the same order of accuracy as the underlying integrators. For applications, some concrete mass-momentum-energy-preserving integrators are derived for the KdV equation.
