Search Results (112)

This paper studies a nonlinear Schrödinger equation with a variable-order time-fractional derivative. Because classical L1 and L1-2 schemes are not directly applicable to variable-order fractional operators, an improved L1-2 discretization with dynamically updated convolution weights is developed based on the Coimbra-type definition, in which the fractional order is evaluated at the current time. By combining the proposed temporal approximation with the Galerkin finite element method for spatial discretization and a linearized extrapolation technique for the nonlinear terms, a fully discrete numerical scheme is constructed. The unconditional stability of the scheme is rigorously proven, and optimal error estimates are established under a mild time step restriction. Numerical experiments are presented to confirm the theoretical results and to demonstrate the effectiveness of the method in capturing the influence of time-dependent memory effects on wave propagation. A key numerical observation is that stronger memory effects may suppress wave packet evolution, which is qualitatively reminiscent of a Zeno-like inhibition phenomenon. Full article

This article proposes a novel time-space two-grid high-order compact difference scheme for the one-dimensional nonlinear Schrödinger equation subject to Dirichlet boundary conditions. In comparison with the fully nonlinear compact difference scheme, the proposed methodology combines a small-scale nonlinear fourth-order compact difference algorithm on a time-space coarse grid and a large-scale linearized correction compact difference algorithm on a fine grid. In contrast to the time two-grid compact difference method, the proposed scheme applies the two-grid technique in both the spatial and temporal domains, thereby further improving computational efficiency. Solutions from the coarse grid are projected onto the fine grid via a temporally linear and spatially cubic Lagrange interpolation operator. Unconditional stability and optimal convergence rates, which are fourth-order in space and second-order in time, are proven in both the discrete $L^2$ and $L^{\infty }$ norms, without any constraints on the grid ratio. In addition to the standard techniques of the energy method, a discrete Sobolev inequality and an a priori error estimate are employed to demonstrate stability and high-order convergence. Finally, the theoretical results are validated through numerical experiments, which confirm the robustness and reliability of the proposed approach. A single-soliton experiment demonstrates that, compared with the fully nonlinear compact difference scheme, the proposed method achieves a significant reduction in CPU time while maintaining a comparable level of accuracy. Additional experiments further illustrate the algorithm’s effectiveness in simulating two-soliton interactions and soliton birth. These findings establish the proposed scheme as a highly efficient alternative to conventional nonlinear approaches. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

This paper presents a computational study of wave–bathymetry and wave–structure interaction problems using advanced numerical techniques based on high-fidelity, two-phase Navier–Stokes (TpNS) flow and reduced-order, fully nonlinear potential flow models. For high-fidelity simulations, the TpNS equations are discretized using the finite-element method, with free-surface evolution captured through a hybrid level-set (LS) and volume-of-fluid (VOF) formulation. A monolithic, phase-conservative LS equation is introduced to mitigate mass loss and interface smearing, combined with a semi-implicit projection scheme. Hydrodynamic forces are resolved using a high-order, phase-resolving cut finite-element method (CutFEM), which enables the representation of complex solid geometries within a fixed background mesh. An equivalent polynomial of Heaviside and Dirac distributions ensures accurate evaluation of surface and volume integrals. Hence, no explicit generation of cut cell meshes, adaptive quadrature, or local refinement is required. For reduced-order modeling, a fast regularized boundary integral method (RBIM) is employed to solve the fully nonlinear potential flow. Singular and near-singular integrals are treated using a subtract-and-addition technique based on auxiliary functions derived from Stokes’ theorem, allowing direct application of high-order quadrature without conventional boundary element discretization. An arbitrary Lagrangian–Eulerian (ALE) formulation is adopted to enforce free-surface boundary conditions while avoiding excessive mesh distortion. The proposed approaches are applied to investigate highly nonlinear wave transformation over complex bathymetry and wave-induced dynamics of floating structures, including eddy-making damping effects. Numerical results are validated against experimental measurements. These two modeling approaches represent complementary levels of physical fidelity and computational efficiency, and their systematic comparison clarifies the trade-offs between computational accuracy, efficiency, and cost for practical marine problems. Full article

This paper considers the numerical approximation of the time fractional Allen–Cahn equation with initial and periodic boundary conditions, and a linear fully discrete scheme is constructed with the finite difference method in time and the Fourier spectral method in space. Based on a temporal–spatial error splitting argument, the boundedness of numerical solutions in the $L^{\infty }$ norm is rigorously proved and the unconditional convergence of the proposed scheme is obtained. Numerical examples illustrate the theoretical results. Full article

In this paper, the nonlinear time-fractional convection-diffusion equations are solved by the Legendre spectral method. The Caputo time-fractional derivative is discretized by the $L2$–$1_{\sigma }$ scheme. A priori estimates of the fully discrete scheme are derived, and the existence and uniqueness of the numerical solution are analyzed. It is rigorously proved that the fully discrete scheme is unconditionally stable, and the convergence order of the numerical scheme is $O(N^{1-m}+{\tau }^2)$. Finally, numerical results are presented to verify the theoretical analysis. Full article

This paper investigates the conservation of mass and energy in the space-fractional Klein-Gordon-Schrödinger system with fractional Laplacian operators. Firstly, the invariant energy quadratization method is applied to transform the original system into an equivalent form. For spatial discretization, Fourier spectral methods are employed, yielding a semi-discrete scheme. Subsequently, an invariant energy quadratization Runge-Kutta approach is used for temporal discretization, resulting in a fully discrete scheme. Owing to its diagonally implicit structure, the proposed scheme is both highly accurate and efficient while preserving mass and energy exactly. Numerical experiments are conducted to verify the accuracy and conservation properties of the method. Full article

We investigate the discrete energy behavior and long-time stability of a second-order Crank–Nicolson mixed finite element discretization for the shallow water equations with nonlinear bottom friction. The method combines a compatible ${\mathrm{BDM}}_1$–${\mathrm{DG}}_0$ spatial approximation with a skew-symmetric formulation of the advective terms and a midpoint treatment of dissipative source terms. At the fully discrete level, we derive a precise mechanical energy identity showing that the scheme is energy-consistent;the discrete energy satisfies a balance law consisting of a nonnegative frictional dissipation term and a higher-order midpoint defect of the order $\mathcal{O}(\Delta t^3)$. Although the method is not unconditionally energy-dissipative, we prove that strict Lyapunov decay holds under a mild CFL-type restriction on the time step. Furthermore, we establish uniform long-time boundedness of the discrete energy and asymptotic recovery of the continuous dissipation law as $\Delta t\to 0$. We also analyze the interaction between nonlinear solver tolerances and energy diagnostics, showing that the observed positive energy increments are controlled, non-accumulating, and intrinsic to the midpoint quadrature structure rather than solver artifacts. The scheme is proven to be precisely well balanced for lake-at-rest equilibria, including nonlinear bottom friction. Comprehensive numerical experiments confirm second-order temporal accuracy, robustness under friction, asymptotic monotonicity under time step refinement, and strict equilibrium preservation. The results provide a rigorous energy-diagnostic framework clarifying when Crank–Nicolson schemes are physically reliable despite the absence of unconditional discrete dissipation. Full article

We propose a fast L2-1_σ finite element method for solving the time fractional Keller–Segel equations with a Caputo fractional derivative of $\alpha \in (0,1)$. Firstly, the fast L2-1_σ scheme on the graded mesh is used to discretize the time fractional derivative. This approach relies on the sum of exponentials (SOE) skill to speed up the convolution kernel. Thus, we overcome the computational cost caused by the nonlocality of fractional derivatives. Then, by combining finite element discretization in spatial direction, a fully implicit numerical scheme is derived. Subsequently, we establish the stability and an $\alpha$-robust error analysis of the fully discrete scheme. Finally, we present some numerical examples to demonstrate the correctness of our theoretical results. Full article

Designing structure-preserving numerical schemes for the generalized Poisson-Nernst-Planck (PNP) system is challenging due to its inherent strong nonlinearity and coupling. In this paper, we propose a class of efficient, unconditional energy-stable schemes based on the Stabilized Scalar Auxiliary Variable (S-SAV) framework combined with the finite element method. We construct both first-order (BE-S-SAV) and second-order (BDF2-S-SAV) fully discrete schemes. A distinguishing feature of our approach is the use of a linear decomposition strategy, which decouples the complex nonlinear system into a sequence of linear, constant-coefficient elliptic equations at each time step. This significantly reduces computational complexity by avoiding expensive nonlinear iterations. We provide rigorous theoretical proofs demonstrating that the proposed schemes are unconditionally energy stable and strictly preserve mass conservation. Numerical experiments satisfy the theoretical analysis, confirming optimal convergence rates and demonstrating robust preservation of mass conservation and modified energy stability in the tested regimes. Full article

Multi-access edge computing (MEC) has been widely recognized as a promising solution for alleviating the computational burden on edge devices, particularly in supporting fast and real-time processing of resource-intensive applications. In this paper, we propose a decentralized offloading decision strategy based on multi-agent deep reinforcement learning (MADRL), aiming to minimize the overall task completion latency experienced by edge devices. Our proposed scheme adopts a grant-free access mechanism during the initialization of offloading in a fully decentralized manner, which serves as the key feature of our strategy. As a result, determining the optimal offloading factor becomes significantly more challenging due to the simultaneous access attempts from multiple edge devices. To resolve this problem, we consider a discrete action space-based deep reinforcement learning (DRL) approach, termed deep Q network (DQN), to enable each edge device to learn a decentralized computation offloading policy based solely on its local observation without requiring global network information. In our design, each edge device dynamically adjusts its offloading factor according to its observed channel state and the number of active users, thereby balancing local and remote computation loads adaptively. Furthermore, the proposed MADRL-based framework jointly accounts for user association and offloading decision optimization to mitigate access collisions and computation bottlenecks in a multi-user environment. We perform extensive computer simulations using MATLAB R2023b to evaluate the performance of the proposed strategy, focusing on the task completion latency under various system configurations. The numerical results demonstrate that our proposed strategy effectively reduces the overall task completion latency and achieves faster convergence of learning performance compared with conventional schemes, confirming the efficiency and scalability of the proposed decentralized approach. Full article

This paper presents a reduced-order Legendre–Galerkin extrapolation (ROLGE) method combined with the scalar auxiliary variable (SAV) approach (ROLGE-SAV) to numerically solve the time-fractional Allen–Cahn equation (tFAC). First, the nonlinear term is linearized via the SAV method, and the linearized system derived from this SAV-based linearization is time-discretized using the shifted fractional trapezoidal rule (SFTR), resulting in a semi-discrete unconditionally stable scheme (SFTR-SAV). The scheme is then fully discretized by incorporating Legendre–Galerkin (LG) spatial discretization. To enhance computational efficiency, a proper orthogonal decomposition (POD) basis is constructed from a small set of snapshots of the fully discrete solutions on an initial short time interval. A reduced-order LG extrapolation SFTR-SAV model (ROLGE-SFTR-SAV) is then implemented over a subsequent extended time interval, thereby avoiding redundant computations. Theoretical analysis establishes the stability of the reduced-order scheme and provides its error estimates. Numerical experiments validate the effectiveness of the proposed method and the correctness of the theoretical results. Full article

We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that the scheme can be unconditionally energy-stable. In particular, the equation becomes the space-fractional Schrödinger equation. Then, the scheme can keep both the discrete mass and energy conserved. Moreover, convergence of the scheme is obtained. Numerical experiments are performed to confirm the theoretical results. Full article

This paper develops a fully discrete finite element scheme for the fractional wave equation with order $\alpha \in (1,2)$, whose solution typically exhibits a weak singularity near the initial time $t=0$. By introducing an auxiliary variable, we first reformulate the fractional wave problem into an equivalent coupled system of two fractional equations. The resulting coupled system is then discretized using the nonuniform Alikhanov formula in time and the standard finite element method on triangular meshes in space. Through rigorous analysis, we establish a pointwise-in-time error estimate for the proposed scheme in the $H^1$ semi-norm. A key advantage of the proposed methodology is its ability to employ a sparser mesh near the initial time to achieve optimal convergence of local errors. In particular, our analysis shows that away from the initial time, the local rate of convergence reaches $O\left(N^{-2}\right)$ in time for $r\ge 2$. Finally, numerical experiments are given to verify the sharpness of the theoretical convergence rates. Full article

In this work, we propose a nonlinear fractional partial differential equation model incorporating a Caputo fractional derivative in time, a second-order spatial derivative, and a nonlinear Fredholm integral term. This model accounts for memory effects, anomalous diffusion, and nonlocal interactions, offering a more realistic description of complex transport phenomena compared to classical integer-order models. To solve the model numerically, we develop a fully discrete scheme that combines Lagrange interpolation-based approximation for the Caputo fractional derivative in time with central difference discretization for the spatial derivative. This approach ensures accuracy and flexibility in handling both the fractional derivative and the nonlinear integral term. A comprehensive convergence and stability analysis is conducted, establishing second-order accuracy in space and nearly second-order accuracy in time. Rigorous error estimates confirm the reliability and robustness of the proposed scheme for practical computations. Finally, a numerical example with a known exact solution is solved to validate the method. Errors are computed in both the $L_2$ and maximum norms, and the temporal and spatial convergence orders are verified. The results, summarized in tables, demonstrate the effectiveness of the fully discrete scheme and underscore the practical utility of the proposed fractional model in complex physical and engineering systems. Full article