In this research work, we investigate a residual-based approach for a posteriori error estimation within the framework of the weak Galerkin finite element method (WGFEM), focusing on singularly perturbed reaction-diffusion problems in the energy norm. Capturing the layer behaviour in the reaction-dominated regime is more complicated compared to the convection-dominated regime, a weighted robust estimator is employed for accurately capturing the layer behaviour for this model problem, which has a reaction-dominated regime. The global reliability and efficiency are verified by deriving both the upper and lower bounds of the proposed estimator. An adaptive WGFEM is developed using this estimator, employing a mesh refinement strategy to effectively resolve the layers for the complicated reaction-dominated regime. Theoretically, the proposed estimator is both reliable and efficient, while numerical results on the unit square and the unit circular domains confirm its effectiveness.
This paper presents the development of a stable, higher-order numerical scheme for solving non-linear generalized time-fractional diffusion equations (GTFDEs). The proposed difference scheme achieves a convergence order of 3-α , 0<α <1 . Spatial discretization is performed using a second-order finite difference operator, and the non-linear term is approximated via higher-order Taylor series expansion. A rigorous stability and convergence analysis is carried out in the L_2 norm. In addition, the numerical stability of the scheme is examined under random noisy perturbations. Numerical experiments on three test problems demonstrate the robustness and efficiency of the proposed method.
This paper focuses on positivity-preserving approximations of a generalized Aït-Sahalia interest rate model with Poisson jumps. In addition to the challenge caused by a drift that blows up at the origin, the highly nonlinear diffusion coefficients and the positivity-preserving requirement, the current study faces essential difficulties due to the presence of the Poisson jumps. By incorporating implicitness in the term α _-1x^-1 and introducing some modification functions f_h and g_h , a novel family of explicit Euler-type schemes is proposed, which is easy to implement and preserves the positivity of the original model unconditionally, i.e., for any time step size h>0 . A mean-square convergence rate of order 1/2 is established for the proposed scheme in both the non-critical and general critical cases. Finally, numerical experiments are provided to confirm the theoretical findings.
Numerical methods are developed for the time-dependent smooth quantum hydrodynamic (QHD) model by using a mixture of hyperbolic, parabolic, and elliptic partial differential equation methods: (i) the underlying hyperbolic gas dynamical part of the transport equations is solved with a third-order weighted essentially non-oscillatory (WENO) method, treating the electric field, scattering, and quantum terms as source terms; (ii) the parabolic heat conduction term is incorporated with the trapezoidal rule/backward difference formula second-order (TRBDF2) method, and (iii) the elliptic Poisson equation is solved with a standard (sparse direct or modern iterative) elliptic solver. For the time-dependent simulations, a regularization of the smooth QHD model equations to prevent an unstable growing mode is implemented. The regularization involves replacing the spatial derivative of the electron density on the right-hand side of the momentum conservation equation by using a quantum generalization of the semiclassical Boltzmann distribution for electron density that captures in a simple way essential effects of quantum tunneling and resonance. Time-dependent simulations of the resonant tunneling diode to steady state using the smooth QHD model are presented, which show realistic negative differential resistance (NDR) (the experimental signal of quantum resonance) in the current-voltage curve. These are the first time-dependent simulations of the smooth QHD model. The simulations match fully quantum mechanical simulations of the resonant tunneling diode much better than any other QHD simulations to date.
In this paper, we present an unconditionally stable numerical scheme for the modified Modica-Mortola (MM) equation with a variable interfacial parameter. The original MM equation with a constant interfacial parameter is derived from a functional having a multiple-well potential. The MM model has a good characteristic that a single function can effectively represent an unlimited number of phases. However, a drawback arises in that the interfacial transition layer widens proportionally to the jump height between phases. To resolve this problem, a novel variable interfacial parameter is introduced to the MM equation so that the interfacial transition layer is approximately constant regardless of any jumps between different phase levels. In this study, we propose an unconditionally stable numerical method for the modified MM equation with a variable interfacial parameter depending on the gradient of the phase-field function. The proposed numerical algorithm consists of the operator splitting scheme with a closed-form solution and the Fourier spectral method. To validate the performance of the proposed numerical method, we conduct a series of computational experiments. The numerical results confirm the superiority of the proposed numerical algorithm.
The resistance distance between two vertices in a connected graph is defined as the effective resistance between them when each edge of the graph is replaced by a unit resistor. The multiset of all pairwise resistance distances in a graph G constitutes its resistance spectrum, denoted RS(G). A graph G is said to be determined by its resistance spectrum (DRS) if, for any graph H satisfying RS(H) = RS(G) , we have H ≅ G . While computational studies reveal that the vast majority of small graphs are DRS, providing rigorous mathematical proofs for specific infinite families remains a challenging and active research area. This paper investigates the resistance spectral determination for four significant families of graphs: barbell graphs, chained silicate networks, Dutch windmill graphs, and caterpillar trees. By employing fundamental principles from electrical network theory—including the series and parallel rules, the cut-vertex principle, and Foster’s theorems—we explicitly analyze the resistance structure of these graphs. We then prove that each family is uniquely DRS. Our results contribute new infinite families of graphs to the growing catalog of those known to be DRS, thereby enhancing our understanding of the expressive power of the resistance spectrum in graph characterization.
In this paper, a second-order backward differentiation formula (BDF2) H^1 -Galerkin mixed finite element method (FEM) is developed for solving the nonlinear Kirchhoff-type equation with a damping term. By introducing a new variable q=∇ u_t+(1+‖∇ u‖ ^2)∇ u , the original hyperbolic equation is transformed into two novel parabolic equations. By means of mathematical induction, the derivative transfer technique, and the technique of recombination for some terms, the superconvergence results with O(h^2+τ ^2) of u in the H^1 -norm and ∇·q in the L^2 -norm are derived in detail. At last, a numerical example is carried out to illustrate the correctness of the theoretical analysis.
This paper addresses two fundamental problems posed by Qi [18] regarding the sufficiency of eigenvalues for the classification of symmetric tensors in the two-dimensional setting. For 2× 2× 2 and 2× 2× 2× 2 complex symmetric tensors, we establish their complete set of equivalence classes via a one-to-one correspondence with the canonical forms of their associated binary cubics and quartics. We then prove that these equivalence classes are uniquely determined by spectral invariants, specifically, the number of eigenpair classes and the multiplicities of zero eigenvalues, over the complex domain. We demonstrate that this classification does not hold in the real domain, where distinct equivalence classes can share identical spectral invariants. Finally, we extend this approach to derive canonical forms and complete classification for complex third- and fourth-order linear partial differential equations (PDEs) in two variables using their bijective relationship to binary forms.
In this paper, we consider a class of difference-of-convex (DC) optimization problems, whose objective function is the difference of a relatively smooth convex function and a continuously convex function. We first propose a novel Bregman-Frank-Wolfe (BFW) algorithm for solving a DC optimization problem. In our algorithm, we mainly use the Bregman distance rather than the Euclidean distance in the selection of the step size. Moreover, we introduce an Adaptive BFW (ABFW) algorithm by adaptively selecting the step-size parameter L_k that corresponds to the relatively smooth information of the objective function. The convergence analysis of the proposed algorithm can be established based on the triangle scaling property. We prove that all accumulation points of BFW and ABFW algorithms are stationary points. Then, we show that the convergence rates of the Frank-Wolfe (FW) gap of BFW and ABFW algorithms are both 𝒪 (k^1-γ/γ) . Finally, some numerical experiments on non-convex quadratic inverse problems are conducted to demonstrate the effectiveness of our algorithms.
For a Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} polynomial, its normal derivatives on the element boundaries are still Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_k$$\end{document} (Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} in three dimensions) polynomials. For a Bell Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} finite element function, its normal derivatives on the element boundaries are Pk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{k-1}$$\end{document} (Qk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{k-1}$$\end{document} in three dimensions) polynomials. We construct a Bell C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>1$$\end{document}-Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} finite element on rectangular meshes in two dimensions and three dimensions for k >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geqslant 4$$\end{document}. We show, with a big reduction from the standard Bogner-Fox-Schmit (BFS) C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>1$$\end{document}-Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} finite element, the C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>1$$\end{document}-Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} Bell finite element retains the optimal order of convergence. Numerical experiments are performed, comparing the new elements with the original elements.
In this paper, we study the influence of the prudence level on the dynamics of a predator-prey model incorporating the additive Allee effect and the fear factor. First of all, we prove the positivity and boundedness of the system solution. Then we investigate the existence and local stability of equilibria and prove the global stability of the positive equilibrium using the Dulac Theorem. Furthermore, we also demonstrate the occurrence of various bifurcations, such as saddle-node, transcritical, and Hopf bifurcations. These theoretical results are proved with numerical simulations. Through numerical simulations, we find the following. (i) An appropriate level of prudence and a weak Allee effect can improve the system stability and promote the coexistence of predators and prey. However, a strong Allee effect may lead to the extinction of predators. (ii) When the level of prudence exceeds a certain threshold, predators will become extinct, but this threshold will increase as the Allee effect level decreases.
Strong ℋ -tensors play a significant role in identifying the positive definiteness of homogeneous polynomials. Motivated by the definition of 𝒩 -scal matrices, this paper first introduces two new subclasses of strong ℋ -tensors, namely 𝒩 -scal tensors and strong 𝒩 -scal tensors. Second, it analyzes the relationships among 𝒩 -scal tensors, strong 𝒩 -scal tensors, and Nekrasov tensors. Finally, as applications, two new methods are proposed to determine the positive definiteness of even-order real symmetric tensors based on 𝒩 -scal tensors and strong 𝒩 -scal tensors, with numerical examples provided to illustrate the results.
This article introduces an algebraic framework for establishing eigenvalue bounds for symmetric positive definite tensors by leveraging intrinsic invariants—specifically, the trace and determinant (resultant). We derive a hierarchy of inequalities via the Arithmetic Mean-Geometric Mean (AM-GM) inequality that yields progressively tighter upper and lower bounds for the tensor’s spectral radius and smallest eigenvalue. A comprehensive comparative analysis demonstrates that our invariant-based approach significantly outperforms classical coordinate-dependent methods such as the Gershgorin circle theorem. We explicitly show that our bounds remain robust and informative in scenarios where Gershgorin bounds fail, particularly for tensors with negative off-diagonal entries (where algebraic cancelations occur) and higher-order tensors (where combinatorial explosion leads to loose estimates). Furthermore, we validate the practical utility of these bounds by applying them to certify the positive definiteness of Lyapunov functions for the stability analysis of nonlinear autonomous systems.
In this paper, we study the time-fractional Cahn-Hilliard (TFCH) equation, which models phase separation processes along with nonlocal memory effects. A fully-discrete numerical scheme is developed using the ultra-weak discontinuous Galerkin (UWDG) method in space with generalized numerical fluxes, coupled with a non-uniform L1 time-stepping scheme. The proposed method is demonstrated to preserve key physical properties of the continuous model, including mass conservation and energy dissipation. We establish the unconditional unique solvability of the fully-discrete scheme using the convex-concave splitting approach and derive stability bounds for the order parameter. An optimal a priori error estimate in the L^2 -norm is established, confirming the convergence of the proposed scheme. Finally, numerical experiments are performed to validate the theoretically predicted convergence order for different choices of numerical fluxes. Numerical simulations confirm the scheme’s accuracy, demonstrating energy dissipation, mass conservation, and phase separation phenomena.
We introduce a nonstaggered central scheme (NCS) for the two-dimensional hyperbolic conservation laws on triangles. The NCS is Riemann-Problem-Solver-Free and is a novel family of central schemes. It has the same simple and clear features as the staggered central scheme. One of the main contributions is that we analyze the convergence property of the NCS on triangular meshes. A dissipation-reducing parameter is constructed to improve the resolution. We introduce a novel method to prove that numerical solutions obtained by the NCS strongly converge to the entropy solution of the two-dimensional scalar conservation law. Several benchmark problems are carried out to verify the convergence and robustness properties. Finally, we show several numerical results of the NCS for the Euler equations on triangles.
This study proposes a high-precision finite element method (FEM) for nonlinear two-dimensional space-time-fractional diffusion equations, combining the Galerkin spatial discretization with the L2- 1_σ temporal scheme. By rigorously analyzing unconditional stability and deriving error estimates, the method achieves spatial convergence of order 2 and temporal convergence of order 3-α , where α∈ (0,1) , validated through numerical experiments under diverse initial conditions. Compared to existing works focusing on linear systems or single-term fractional dynamics, this research innovatively extends the L2- 1_σ scheme to complex nonlinear space-time-coupled problems, demonstrating its capability to simultaneously handle nonlinearities and fractional derivatives while maintaining computational efficiency and geometric adaptability.
For convection-diffusion-reaction equations, we are interested in the dynamic diffusion (DD) method, which is free of stabilization parameters and capable of precluding numerical oscillations, and prove the existence and uniqueness of the approximation solution for the DD method by applying the contraction mapping principle. In the energy norm, we propose a residual-type a posteriori error estimator, which is proven to be reliable and efficient, and which is robust when the local Péclet number is not large. Based on the a posteriori estimator, we develop a linearized adaptive DD (LADD) algorithm, and carry out numerical experiments to validate the effectiveness and reliability of the LADD algorithm.
