2026-07-31
Zhenping Li, Xiangtuan Xiong
In this paper, the Cauchy problem of the nonlinear Helmholtz-type equation is discussed. This problem is well known to be severely ill-posed. Compared with linear equations, nonlinear equations are more difficult to deal with because of their lack of linearity. In order to obtain the approximate solution of this nonlinear problem, a combination of the quasi-boundary value method and the quasi-reversibility method is proposed. Using the variable separation method, the approximate solution is equivalent to solving a class of integral equations. The well-posedness of the approximation problem is proved by the Banach fixed-point theorem. Convergence analysis and error estimation are discussed. Since the convergence of error estimates cannot be obtained by the traditional a-priori bound, we introduce a new a-priori bound to obtain the convergence of error estimates, and the Hölder-type error estimate is achieved. Finally, some numerical experiments are given to corroborate the qualitative analysis and show the regularization method works well.
2026-07-31
Zunlin Tan, Xiaoping Xie
This paper considers a parabolic equation involving the Caputo fractional time derivative of order $\gamma\in(0,1)$ and the spectral fractional power, of order $s\in(0,1)$, of a symmetric second order elliptic operator $\mathcal{L}$. By using the Caffarelli–Silvestre extension, the original problem is converted to a quasi-stationary elliptic problem on a semi-infinite cylinder in one more spatial dimension and with a dynamic boundary condition. Based on the solution representation with the Mittag–Leffler function and the eigenpairs of the elliptic operator $\mathcal{L}$, some new regularity results are derived. For the temporal semi-discrete scheme using the Alikhanov difference approximation, stability and convergence are established. Furthermore, the first-degree tensor product finite element method is adopted for the spatial discretization to obtain a fully discrete scheme, and an error estimate is derived. Finally, an efficient algorithm based on a generalized eigenvalue problem is applied to solve the matrix system of the full discretization. Numerical examples are provided to verify the theoretical results.
2026-07-31
Heng Cheng, Dongqiong Liang, Fengbin Liu
In this study, by combining the orthonormal basis functions and the traditional moving least-squares (MLS) approximation, thus the new approximation function is established by using the improved moving least-squares (IMLS) approximation, and the corresponding formula derivation is given in Section 2. Afterwards, the equilibrium, geometrical and physical equations of bending problems of plates on elastic foundations are presented respectively, and the equivalent functional of such problems is established with the essential boundary condition imposed by using the penalty method, thus the calculation formula of numerical solution are derived using the improved element-free Galerkin (IEFG) method with IMLS approximation. In numerical examples, we verify the convergence of the IEFG method by increasing the number of nodes. In comparison with the EFG method, the IEFG method converges faster. Moreover, the smaller error and higher calculation speed are obtained by selecting the IEFG method for solving three numerical examples.
2026-07-31
Yiming Du, Mingze Ma, Jinrong Zhang, Hanqi Song, Chao Yan
This paper presents a composite wall function, which can be used for a wide range of the adverse pressure gradient (APG) and is derived from the composite wall-law provided in this study. The composite wall-law combines the characteristics of the log-law and the half-power law under the APG, and it degrades to the log-law under the zero pressure gradient (ZPG). A function relationship between the parameters in the composite wall-law and the dimensionless pressure gradient parameter $\mathcal{p}^+$ is provided based on a series of DNS data for the APG. Finally, a generalized wall function incorporating both the viscous sublayer and the buffer layer is presented. The new wall function can accurately predict the mean velocity of both equilibrium and nonequilibrium flows under the APG. The new composite wall function is validated using three test cases: NACA0012, 2D channel with a bump, and turbulent separation bubble. The results demonstrate the accuracy of the new composite wall function under the APG flows.
2026-07-31
Jianming Wu, Cong Huang, Lili Chen
This paper develops an adaptive-linear-weights, multi-resolution WENO method in the framework of subcell (AMWENO-SC). The features of AMWENO-SC present on that: first, it uses the subcell strategy; second, it only consists of two subreconstructions, where one is first order, the other is higher order and designed by using the CLS; third, it optimizes the smoothness indicator of first order subreconstruction by using the WLS-ENO; finally, it simplifies the adaptive linear weights by using the TENO approach. Compared with original AMWENO, our AMWENO-SC has better compactness, higher resolution and efficiency.
2026-05-31
Tiao Lu, Xingchun Yu, Wenqi Yao
A conservative sharp interface method based on the cut-cell approach is proposed for the simulation of compressible two-medium flows on unstructured triangular meshes. In this method, the governing equations for each medium are solved using a MUSCL-type finite volume scheme. The exact two-medium Riemann problem is solved at the moving interface, while the local Lax–Friedrichs flux is employed along the fixed cell edges. The interface is tracked by solving the advection equation for the level set function, and explicit interface reconstruction is performed at each time step within the cut-cell framework. To address the small cell problem and ensure global conservation, a novel cell-merging algorithm, specifically designed for unstructured triangular meshes, is introduced to ensure that interface cells coincide with the interface. In contrast to existing cut-cell methods used on Cartesian meshes, the complexity of the new algorithm in constructing cut cells is significantly reduced. Additionally, the adaptive mesh refinement technology is employed to efficiently simulate complex flow problems while reducing the computational cost. A series of numerical tests is conducted to demonstrate the conservativeness and accuracy of the proposed method.
2026-05-31
Anh Ha Le, Quan M. Nguyen
In this study, we implement a Crank-Nicolson finite difference scheme to discretize the (2+1)D saturable nonlinear Schr ö dinger equation with general damping. We show the existence and uniqueness of the discrete solution. The boundedness of the discrete mass and energy is established. The error between the exact and discrete solutions is shown to converge at a second-order rate in both time and space, according to the $L^2$ and $H^1$ discrete norms. Moreover, we show that the proposed scheme preserves the mass conservation and energy conservation for the (2+1)D saturable nonlinear Schr ö dinger equation without damping. Numerical simulations are conducted to validate these convergence properties and the conservation laws.
2026-05-31
Lukas Einkemmer, Trung-Hau Hoang, Alexander Ostermann
In this paper, we consider the application of exponential integrators to problems that are advection dominated. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators, depending on the specific setting, either outperform or perform similarly to explicit Runge–Kutta schemes. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.
2026-01-29
Daniel Walsken, Pavel Petrov, Matthias Ehrhardt
In this study, a Fourier-based split-step Padé (SSP) method for solving the parabolic wave equation with applications in guided wave propagation in ocean acoustics is presented. Traditional SSP implementations rely on finite-difference discretizations of the depth-dependent differential operator. This approach limits accuracy in coarse discretizations as well as computational efficiency in dense discretizations, since it does not significantly benefit from parallelization. In contrast, our proposed method replaces finite differences with a spectral representation using the discrete sine transform (DST). This enables an exact treatment of the vertical operator under homogeneous boundary conditions. For non-constant sound speed profiles, we use a Neumann series expansion to treat inhomogeneities as perturbations. Numerical experiments demonstrate the method's accuracy in range-independent and range-dependent scenarios, including propagation in deep ocean with Munk profile and in the presence of a parameterized synoptic eddy. Compared to finite-difference SSP methods, the Fourier-based approach achieves higher accuracy with fewer depth discretization points and avoids the resolution bottleneck associated with sharp field features, making it well-suited for large-scale, high-frequency wave propagation problems in ocean environments.
2026-01-12
Debabrata Singh, Madan Mohan Panja
The present work is concerned with developing a numerical scheme (viz., sinc-Galerkin method) based on sinc functions for the investigation of various 1D peridynamic models. Four different peridynamic models corresponding to four micromodulus functions have been investigated here. The $a$ $posteriori$ errors in the approximation has been estimated and compared with the absolute errors whenever exact solutions are available. The results of the investigation ensure that the proposed scheme may be quite effective and useful for the analysis of 1D peridynamic models.
2025-12-21
Dongdong Hu, Haorong Huang, Yayun Fu, Minghua Chen
In this paper, we propose and analyze a class of linearly implicit energy stable scheme for the nonlinear time-fractional Allen-Cahn equation with a general potential function, where the temporal and spatial derivatives are approximated by the variable-step L1 method and the central difference method, respectively. The proposed scheme is proved to be unconditionally energy-dissipation-preserving and maximum-bound-principle-preserving in discrete settings with the help of the discrete orthogonal convolution technique. Thanks to the maximum-normal boundness of numerical solution and the discrete fractional Grönwall inequality, we obtain the convergence in $L^2$ norm of the proposed scheme. In practical computation, we utilize the graded mesh and the adaptive mesh to simulate the time-fractional Allen-Cahn equation to capture the multi-scale behaviors and enhance computational efficiency. Extensive numerical comparisons are provided to verify the correctness and efficiency of the proposed scheme in long-time computations.
2025-09-28
Yinghao Chen, Muzhou Hou, Jingying Hu, Yang Xiao, Jinyong Ying, Ting Zhang
This paper presents ALM-PINN, an adaptive physical informed neural network algorithm optimized by Levenberg-Marquardt. ALM-PINN is tailored to overcome challenges for solving singular perturbation problems (SPP). Traditional neural network-based solvers reframe solving differential equations task as a multi-objective optimization problem involving residual or Ritz error. However, significant disparities in the magnitudes of loss functions and their gradients frequency result in suboptimal training and convergence challenges. Addressing these issues, ALM-PINN introduces a learnable parameter for the perturbation parameter and constructs a two-terms loss function. The first loss term emphasizes approximating the governing equation, while the second term minimizes the difference between perturbation and learnable parameters. This adaptive learning strategy not only mitigates convergence issues in directly solving SPP but also alleviates the computational burden with asymptotic iteration from a large initial value. For one-dimensional tasks, ALM-PINN enhances training efficiency and reduces complexity by enforcing hard constraints on boundary conditions, streamlining the loss function sub-terms. The efficacy of ALM-PINN is evaluated on five SPPs, demonstrating its ability to capture sharp changes in physical quantities within the boundary layer, even with small perturbation coefficients. Furthermore, ALM-PINN exhibits reduced errors in both $L_ {\infty}$ and $L_2$ norms, coupled with improved convergence speed and stability.