In this paper, we introduce a novel fifth-order weighted compact finite volume scheme that integrates the concept of adaptive order weighted essentially non-oscillatory (WENO) schemes into existing linear compact schemes. Our proposed method employs a new stencil selection criterion for the smoothness indicator, resulting in a convex combination of one fifth-order linear compact reconstruction and two third-order linear reconstructions. This innovative combination preserves the high-resolution characteristics of compact schemes while effectively capturing discontinuities with the accuracy of WENO methods. To demonstrate the efficiency of our scheme, we applied it to solve a range of nonlinear scalar equations and systems of Euler equations in both one- and two-dimensional spaces. The results show that the intended fifth-order accuracy is maintained in smooth regions, and the resolution near shocks or discontinuities is significantly enhanced without any numerical oscillations. Additionally, the computational cost associated with using the same stencils is reduced.
In this paper, we investigate the solutions to the dual quaternion matrix equation , including the general solution, the solution containing only the standard part, and the solution containing only the dual part. First, we present the real representation matrix of the dual quaternion matrix and analyze the properties of its vector operator. Then we transform the dual quaternion matrix equation into a real linear system and give the equivalent condition for the existence of the solutions to the dual quaternion matrix equation , which includes the general solution, the solution containing only the standard part, and the solution containing only the dual part. Furthermore, we obtain the expressions of these solutions and the minimal -norm solutions to the dual quaternion matrix equation . Finally, we propose the algorithms and conduct numerical experiments to verify the validity of our method.
The goal of optimal control is to determine a sequence of inputs for maximizing or minimizing a given performance criterion subject to the dynamics and constraints of the system under observation. This work introduces Control Physics-Informed Neural Networks (PINNs), which simultaneously learn both the system states and the optimal control signal in a single-stage framework that leverages the system’s underlying physical laws. While prior approaches often follow a two-stage process-modeling, the system first and then devising its control—the presented novel framework embeds the necessary optimality conditions directly into the network architecture and loss function. We demonstrate the effectiveness of the novel methodology by solving various open-loop optimal control problems governed by analytical, one-dimensional, and two-dimensional partial differential equations (PDEs).
The very large scale integration (VLSI) placement problem aims to position electronic cells within a circuit region, optimizing objectives such as wire length and ensuring no cell overlaps. Analytical methods frame the placement problem as a constrained optimization problem and solve its smoothed or relaxed version using optimization methods. In this paper, we shall solve the placement problem without employing any smoothing or relaxation technique. We propose an efficient, smoothing-free analytical method for solving the VLSI placement problem. The basic idea is to utilize the nonsmooth proximal alternating direction method (PADM) to deal with the difficulties associated with the nonsmoothness and constraint complexity of the VLSI placement problem. The global convergence of the method is established. Numerical experiments are conducted on the GSRC circuits, which demonstrate the usefulness of the method.
Various statistical properties of the exponentiated Weibull exponential (EWE) distribution including quantile and hazard rate functions, skewness, kurtosis, order statistics, and entropies are investigated. The parameters are estimated by the maximum likelihood estimation (MLE) method. The flexibility and behaviour of the estimators were studied through a simulation. The empirical flexibility of the presented distribution was examined by means of real-life data. It was observed that our distribution serves as a viable alternative model to existing probability densities in the literature for the analysis of lifetime data.
Distributed-order fractional diffusion equations (DO-FDEs) are crucial for modeling complex processes in heterogeneous and anomalous systems. Unfortunately, they encounter significant analytical and computational challenges. This study introduces a novel numerical framework that extends high-order approximation formulas to accommodate the distributed-order fractional derivative. The framework achieves a temporal convergence order of , where is the upper bound of the integral in the distributed-order derivative. The proposed scheme using the finite element method (FEM) in the spatial direction, offers an accurate approach for solving DO-FDEs. We examine stability and convergence analyses to validate the method’s applicability. Additionally, numerical experiments confirm theoretical results.
In this article, we propose and analyze a least-squares-based weak Galerkin finite-element method (WG-FEM) for solving the indefinite time-harmonic Maxwell’s equations in The super-convergence of order one for the discrete -like norm has been established. Numerical simulations show that the approximate solutions converge to the exact solutions with optimal rates in the norm on hybrid meshes. In addition, this method is shown to be absolutely stable under low regularity requirements with high wave numbers.
A line digraph L(D) of a directed multigraph has as its vertex-set being A(D), the set of arcs of D, where (a, b) is an arc of L(D) if and only if there are vertices u, v, and w in D such that and are in A(D). In this paper, we obtain sufficient and necessary conditions for a line digraph L(D) to be supereulerian and to have a spanning trail, respectively, in terms of certain types of path-cycle covers of D. These results will be applied to show each of the following for a digraph D. (i) If , then L(D) is supereulerian. The lower bound on |A(D)| is best possible in the sense that there exists an infinite family of digraphs each of which satisfies without a supereulerian line digraph. (ii) There exists a well-characterized digraph family such that if D is strong with , then L(D) is supereulerian if and only if .
Recently, Lv and Miao (Appl Math Lett 154: 109109, 2024) presented a new inexact fixed point iteration method which is based on an inexact fixed point iteration method for solving the tensor absolute value equation (TAVE). In this paper, we give new convergent conditions of the inexact fixed point iteration method and the new inexact fixed point iteration method, respectively.
Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of for generating sets that contain a matrix whose minimal polynomial has a degree exceeding , where denotes the order of the matrix. Compared to the bound provided in Theorem 3.1 of Guterman et al. (Linear Algebra Appl 543: 234–250, 2018), this result reduces the constraints on the Jordan canonical forms. In addition, it is demonstrated that the bound holds when the generating set contains a matrix with a minimal polynomial of degree satisfying . The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.
In this paper, we present an ultra-weak discontinuous Galerkin (UWDG) method that employs H(div)-conforming spaces for solving incompressible flows, ensuring exact divergence-free solutions. Leveraging the exact divergence-free property and the continuity of the normal velocity at cell interfaces, the convection term is formulated in conservative form and discretized as the DG method for conservation laws. To achieve a high-order time discretization, we first implement a predictor-corrector procedure utilizing either one-step or multi-step methods. Then, we integrate the spectral deferred correction scheme to develop a linear semi-implicit high-order fully discrete scheme. Numerical experiments demonstrate the efficiency and accuracy of this high-order method.
The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. The latter are shared between neighboring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approximations to the spatial derivatives which are used in the point value update. The averages are updated using a quadrature of the flux and making use of the point values as quadrature points. The integration in time employs standard Runge-Kutta methods. We show that this generalization of the Active Flux method in two and three spatial dimensions is stationarity-preserving for linear acoustics on Cartesian grids, and present an analysis of numerical diffusion and stability.
Compared to multi-layer neural networks based on nonlinear activation functions, single-layer neural networks that use smooth linear functions as activation functions can quickly numerically solve time-fractional partial differential equations (TFPDEs) with time derivative of order . However, due to the weak singularity of the solution which is not smooth near the initial time, resulting in significant numerical errors when using neural networks based on smooth functions as activation functions. In this paper, we develop a decomposition method to reduce the singularity of the initial value, and theoretically demonstrate that this method can reduce the singularity. Then, a new single-layer neural network and a fast training method are proposed based on the decomposition method and the smooth linear activation function. Numerical experiments have shown that this method reduces errors and improves computational speed.
In this work, we revisit two modified L1 interpolation approximations to the values of the Caputo fractional derivative of order at the midpoints of the mesh. These approximations are distinct from the classical L1 formula which typically yields interpolation approximations at the mesh points themselves. Although these two formulae on uniform meshes have been proposed in previous literature (Dong et al. in Commun Appl Math Comput 5(4): 1446–1468, 2023; Osman in Numerical solution methods for fractional partial differential equations. PhD thesis, University of Southern Queensland, 2017), given their practical applications in solving fractional differential systems with possible weak singularities and the inherent simplicity of the L1-type formulae compared to the existing higher-order numerical formulae such as the L2- formula and L2-type formulae, it is more essential to consider the corresponding results on non-uniform meshes. To this end, two modified L1 interpolation formulae on non-uniform meshes, subsequently referred to as Formula I and Formula II, respectively, are explored in the current work. On the basis of deriving the formulae, the properties of coefficients in the formulae and the truncation errors under the graded mesh are analyzed in detail. It is worth noting that Formula I is reduced to the classical central difference quotient formula as , which is superior to Formula II and the classical L1 formula. Furthermore, we present several numerical examples to demonstrate the efficacy of these two formulae, as well as the effectiveness of their applications in solving the fractional sub-diffusion equation. The unique solvability and stability of the resultant schemes with respect to the initial values are briefly mentioned. As far as we know, there has been hardly any previous systematic investigation for this type of formulae on non-uniform meshes.
Tensors (hypermatrices) are multidimensional analogs of matrices. In the last few years, the tensor splitting problem has attracted much attention and has been studied extensively, from theory to solution methods and applications. This work, with its two parts, aims to contribute to reviewing the state-of-the-art studies for the tensor splittings and iterative methods for solving multilinear systems (tensor equations). In the first part of this paper, some new comparison theorems for two regular and weak regular splittings of real tensors are derived. These theorems are extensions of the classical comparison theorems of splittings of matrices. We show that, under certain conditions, the iterates implied for solving the multilinear system are monotone. We generalize theorems about H-splittings from matrices to tensors. These and comparison theorems for two regular and weak regular splittings can be easily extended for complex tensors. In addition, we discuss triangular splitting, -splitting, -splitting, -compatible splitting, and direct splitting of tensors. Some iterative methods and associated comparison theorems for solving the multilinear system , when is an -tensor, and a fast and smooth iterative method to solve with the symmetric coefficient tensor are given in the second part.
Convection-diffusion-reaction equations are a class of second-order partial differential equations (PDEs) widely used to model phenomena involving the change of concentration/population of one or more substances/species distributed in space. Understanding and preserving their stability properties in numerical simulations is crucial for accurate predictions, system analysis, and decision-making. This work focuses on the development of a comprehensive numerical framework for a class of convection-diffusion-reaction systems with a dissipative Lyapunov (or entropy or free energy) functional, . This non-increasing Lyapunov functional is the driving quantity of the stability and properties of the system. We introduce a systematic methodology for constructing discretizations that mimic the stability analysis of the continuous model using Lyapunov’s direct method-type approach. The spatial algorithms are based on collocated discontinuous Galerkin (DG) methods with the summation-by-parts (SBP) property and the simultaneous approximation term (SAT) approach for imposing interface coupling and boundary conditions. Relaxation Runge-Kutta schemes are used to integrate in time and achieve fully discrete Lyapunov consistency. To verify the properties of the new schemes, we numerically solve a system of convection-diffusion-reaction PDEs governing the dynamic evolution of monomer and dimer concentrations during the dimerization process. Numerical results demonstrated the accuracy and consistency of the proposed discretizations. The new framework can enable further advancements in the analysis, control, and understanding of general convection-diffusion-reaction systems.