Advances in Applied Mathematics and Mechanics
Advances in Applied Mathematics and Mechanics (AAMM) provides a fast communication platform among researchers using mathematics as a tool for solving problems in mechanics and engineering, with particular emphasis in the integration of theory and applications. To cover as wide audiences as possible, abstract or axiomatic mathematics is not encouraged. Innovative numerical analysis, numerical methods, and interdisciplinary applications are particularly welcome.
A First-Order Linear Energy Stable Scheme for the Cahn–Hilliard Equation with Dynamic Boundary Conditions Under the Effect of Hyperbolic Relaxation
In this paper we focus on the Cahn–Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two stabilization terms, we have constructed a first-order temporal accuracy numerical scheme, which is linear and energy stable. Then we prove that the scheme is of first-order in time by the error estimates. At last we present comprehensive numerical results to validate the the temporal convergence and the energy stability of such scheme. Moreover, we present the differences of the numerical results with and without the hyperbolic terms, which show that the hyperbolic terms can help the total energy decreasing slowly.
Copyright (c) 2025 Advances in Applied Mathematics and Mechanics
High-Order Unconditionally Energy-Stable Decoupled Method for Phase Field Modeling of Pitting Corrosion with Adaptive Implementation
Pitting corrosion, recognized as one of the most catastrophic forms of localized corrosion, frequently leads to premature structural failures. A major computational challenge is the development of accurate and efficient numerical methods that can capture both the initiation and the nonlinear evolution of corrosion pits. In this work, we propose a third-order, decoupled, unconditionally energy-stable implicit-explicit (IMEX) Runge-Kutta scheme for a phase-field model of pitting corrosion, which is formulated as a gradient flow of a coupled free energy functional involving a phase field variable and a normalized ion concentration. The proposed scheme is rigorously proven to satisfy a discrete energy dissipation law, regardless of time step size. To further enhance computational performance, we integrate an adaptive mesh refinement and dynamic time-stepping strategy, enabling efficient resolution of interface dynamics. Comprehensive numerical experiments validate the high accuracy, unconditional energy stability, and effectiveness of adaptive implementation in simulating the complex spatio-temporal evolution of pitting corrosion.
A Spectral Split-Step Padé Method for Guided Wave Propagation
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.
Copyright (c) 2025 Advances in Applied Mathematics and Mechanics https://creativecommons.org/licenses/by-nc-nd/4.0
Explicit Form of Simplified Grad’s 13 Moments Distribution Function-Based Moment Gas Kinetic Solver with Unstructured Meshes for the Multiscale Rarefied Flow
It is essential to efficiently solve multiscale flows covering the continuum regime to the rarefied regime. The explicit form of Grad’s 13-moment distribution function-based moment gas kinetic solver (G13-MGKS) has been proposed in our previous work [Comput. Math. Appl., 137 (2023), pp. 112–125], which demonstrates the potential for efficiently simulating continuum flows accurately and presenting reasonable predictions for rarefied flows at moderate Knudsen numbers on structured meshes. To further extend the solver’s applicability to unstructured meshes, we propose the simplified version of Grad’s 13 moments13-moment distribution function-based moment gas kinetic solver (SG13-MGKS) with an explicit form of the numerical flux in the present paper. The Shakhov collision model has been adopted and validated within the framework of SG13-MGKS to ensure the correct Prandtl number in the simulation. Additionally, a simplified treatment for the numerical fluxes has been adopted to minimize the need for complex calculations of the gradient of integral coefficients. The performance of SG13-MGKS has been evaluated in numerical cases of Couette flow with temperature differences, flow passing through the airfoil, and pressure-driven flow in a variable-diameter circular pipe. Our results demonstrate that SG13-MGKS can achieve reasonably accurate computational results at Knudsen numbers below 0.2. Benefiting from the avoidance of discretization in velocity space, G13-MGKS is able to be two orders of magnitude faster compared to the conventional discrete velocity method. Furthermore, the simplified form of numerical fluxes and the fewer gradients of integration coefficients enable the performance of SG13-MGKS on unstructured grids with a saving of about 4 times the computation time and 3 times the memory cost compared to the previous version of G13-MGKS.
An Efficient Numerical Scheme Based on Sinc Functions for the Analysis of 1D Peridynamic Models
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.
A Three-Dimensional Mass-Conserved Multiphase Lattice Boltzmann Flux Solver for Incompressible Flows with Large Density and Viscosity Ratios
Multiphase lattice Boltzmann flux solver (MLBFS) uses the finite volume method to solve Navier-Stokes (NS) and Cahn-Hilliard (CH) equations. However, instead of macroscopic fluxes, the corresponding mesoscopic fluxes, defined using a local lattice Boltzmann method, are evaluated at cell faces. Since the development of the original three-dimensional (3D) MLBFS [16], several improvements have been made in it, including modification of CH equation to conserve the mass, determination of CH mesoscopic fluxes to eliminate the weighted essentially non-oscillatory scheme, and simplification of mesoscopic fluxes to improve the computational efficiency. However, these improvements have been implemented and studied in two-dimensional or axisymmetric MLBFS. In this study, MLBFS (based on the mentioned improvements) has been extended to 3D flows and applied to simulate three incompressible multiphase benchmark cases with large density and viscosity ratios up to 1000 and 100, respectively. The results of benchmark cases (Laplace law, bubble rising, and drop impact on a dry surface) agree well with previous credential data. Our simulations show the original CH equation considerably reduces the bubble/drop mass, while the modified CH equation conserves it completely. According to results, using the simplified mesoscopic fluxes (instead of the original ones) saves about 13% of computational time.
Two $P_4$ Nonconforming Finite Elements for the Biharmonic Equation on Rectangular Meshes
In this paper, two new third order nonconforming finite element methods (NFEMs) on the rectangular grid are proposed for the biharmonic problem. Such finite elements are constructed by enriching 9 high-order polynomial bubbles to the $P_4$ polynomial space on each rectangle, so that the extra degrees of freedom can enforce the required sub-continuity of the finite element on the four edges. We prove that the methods are well-defined, have unique solution and converge at $\mathcal{O}(h^3)$ in the $H^2$ norm and $\mathcal{O}(h^5)$ in $L^2$ norm. The numerical results show that the new elements are very efficient.
Unconditionally Energy-Dissipation- and Maximum-Bound-Principle-Preserving Scheme for the Time-Fractional Allen-Cahn Equation
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.
Local Interaction Simulation Approach for the Acoustic Wave Equation with Perfectly Matched Layer
The simulation of the acoustic wave equation is essential for a wide range of applications, but the complexity of the propagation medium presents significant challenges. These include addressing the infinite computational domain, which can be managed using absorbing boundary conditions, and handling interface conditions between different media. In this paper, we utilize the Local Interaction Simulation Approach (LISA) to simulate acoustic wave propagation, which is designed to accommodate complex interface geometries on a regular grid. In engineering applications, absorbing boundary conditions in LISA are typically implemented using the ALID method, which introduces a damping term that increases in the absorbing layer. Here, we propose integrating the Perfectly Matched Layer (PML) with LISA, a novel combination that has not been previously explored in the literature. We demonstrate that, despite the added complexity of PML due to its requirement for multiple auxiliary equations within the absorbing layer, it delivers superior performance and outperforms the ALID method in terms of effectiveness.
local interaction simulation approach (LISA)
New Regularization and Error Estimate for the Cauchy Problem of the Nonlinear Helmholtz-Type Equation
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.
Barycentric Interpolation Collocation Methods Based on the Crank-Nicolson Scheme for Nonlinear Parabolic Optimal Control Problems
In this paper, two numerical schemes for nonlinear optimal control problems governed by parabolic equations are presented to deal with the challenge arising from the mutually coupled state and co-state variables in numerical simulation. Using Lagrangian multipliers, the continuous optimality system is derived, which consists of state and co-state equations, coupled with an optimality condition. To achieve higher spatial accuracy, meshless and high-precision barycentric interpolation collocation methods are applied. Fully discrete collocation approximation schemes are presented, utilizing the Crank-Nicolson scheme in time and the Newton linearization for the nonlinear term. To avoid solving the large coupled scheme directly, a classical iterative method is adopted. Furthermore, we provide consistency analyses of semi-discretized schemes in space, as well as nonlinear fully discretized schemes based on approximation properties of collocation methods. Finally, several numerical experiments are conducted to validate the efficiency of our proposed methods. Comparisons with a classical finite difference method indicate that the proposed collocation schemes offer superior accuracy while requiring fewer nodes.
Machine Learning Based Optimization of Tube Geometry for Capillary Rise
The phenomenon of capillary rise is commonly observed in both nature and engineering fields. There has been growing interest in the study of the geometry effect on the capillary rising. In this work, we investigate the effect of tube geometry on the capillary rise, and explore the optimal tube geometry that corresponds to the maximum equilibrium height. To this end, we present two machine learning approaches that are able to optimize the tube shape to obtain maximum equilibrium height given a material. Specifically, the first one is based on the Gaussian process regression (GPR) in a purely data-driven manner, and the second one utilizes deep neural networks (DNNs) which are capable of encoding the physical constraint, e.g., the governing equation for the capillary rise in a tube. We apply both methods to study the capillary rise in the uniform cylindrical, trigonometric-shaped and quadratic-shaped tubes, and then determine the optimal geometries that achieve the maximum equilibrium heights. Our results demonstrate that the both approaches can effectively optimize complex capillary channels corresponding to the equilibrium height. Further, GPR is computationally more efficient than DNNs, while the DNNs with physical constraints are more promising for solving problems in which the tube geometry is parameterized at high dimensions.
High-Order Numerical Approximation and Error Estimation for Fourth-Order Equation in Complex Cylindrical Domain
In this paper, we introduce a high-order numerical method for solving fourth-order equations in an elliptical cylindrical region. Initially, we employ an elliptical cylindrical coordinate transformation to reformulate the fourth-order equation as an equivalent second-order coupled system in the new coordinate system. To overcome the singularity introduced by coordinate transformation, an essential pole condition is derived. The weak form and its discretization are also established. Furthermore, the well-posedness of both the weak solution and its approximate solution have been theoretically investigated. By introducing novel projection operators and demonstrating their approximation properties, we prove the error estimates in conjunction with the approximation results of Fourier series. Finally, several numerical examples are provided to validate the convergence and high accuracy of our proposed schemes.
Multi-Resolution WENO Method with Adaptive Linear Weights Using Subcell Strategy
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.
Analyzing Bending Problems of Plates on Elastic Foundations via Improved Element-Free Galerkin Method
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.
A Fully Discrete Method for Space-Time Fractional Parabolic Problems
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.
Space-time fractional parabolic equation
A Fast Spatial Two-Grid Raviart-Thomas MFE Algorithm for the Nonlinear Time Fractional Diffusion Equations on Temporal Graded Mesh
In this paper, a fast two-grid algorithm is constructed for solving the nonlinear time fractional diffusion equations by using the lowest order Raviart-Thomas mixed finite element (RTMFE) space and the temporal graded mesh. In the algorithm, the Caputo time fractional derivative is discretized by the well-known $L1$ formula on the graded mesh, the spatial domain is divided into coarse and fine grids, then a fast two steps algorithm is proposed by using two-grid computing method. The existence, uniqueness and unconditional stability for the proposed algorithm on the temporal graded mesh are derived in detail. In addition, when the analytical solution satisfies different regularity assumptions, the asymptotically optimal a priori error estimates in spatial direction are obtained on both the temporal uniform and graded meshes, which show that when spatial coarse and fine grid parameters satisfy $H=O(h^{1/2})$, the fast algorithm can obtain the same accuracy as the RTMFE algorithm. Finally, two numerical examples with different regularity conditions are provided to demonstrate the theoretical results.
nonlinear time fractional diffusion equations
Error Analyses of the Single Hidden Layer Neural Network Based on the Extreme Learning Machine for Solving Differential Equations
The differential equations, especially the stiff differential equations with the large eigenvalue and instability of solutions, impose strict limitations on step size for traditional numerical methods. The single hidden layer neural network method based on the extreme learning machine is widely used to solve various differential equations due to its advantages of few parameters and high efficiency, and the optimal estimation of parameters is obtained. In practice, this does not mean that the solutions of differential equations converge well. This paper aims to establish theoretical results for solving non-stiff and stiff differential equations using the single hidden layer neural network based on the extreme learning machine. Under the classical Lipschitz condition and the one-sided Lipschitz condition, we derive convergence results for the approximate solutions of these equations. Numerical experiments validate the theoretical findings, and the results indicate that computational speed can be improved by converting deep neural networks into shallow ones when solving differential equations.
Optimizing Thermal Lattice Boltzmann Method on an MT-3000 Processor with Neo-Heterogeneous Strategies
The Lattice Boltzmann Method (LBM) is a computational fluid dynamics method for simulating fluid flows with the benefits of locality and simplicity, making it ideal for parallel computing and complex flow simulations. This study focuses on developing a specialized Double Distribution Function (DDF) LBM software framework optimized for the MT-3000, a novel heterogeneous processor, to facilitate thermal incompressible flow simulations. To improve LBM’s performance on the complex multi-zone architecture of MT-3000, this paper introduces several innovative strategies. Firstly, a temporal fusion optimization strategy is implemented. This strategy involves postponing the temperature field calculations during time steps, efficiently decreasing the time overhead. Furthermore, we present “Pencil-H”, a novel pipelined algorithm meticulously designed to harness the unique capabilities of the MT-3000, thereby enhancing computational efficiency and communication effectiveness. Additionally, an architecture-aware multi-level parallelization algorithm is proposed, tailored to maximize the computational capabilities of the MT-3000. The effectiveness of these optimization strategies has been thoroughly validated through extensive bench-marking tests. These validations have shown remarkable performance enhancements, including a significant acceleration factor of 32.02X when compared to using 16 CPU cores. Notably, the optimized code demonstrated high-fidelity simulation capabilities for thermal incompressible flows, achieving 61.61% of the theoretical maximum performance defined by the roofline model.
A Conservative Sharp Interface Method for Compressible Two-Medium Flows on Unstructured Triangular Meshes
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.
Should Exponential Integrators Be Used for Advection-Dominated Problems?
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.
A New Composite Wall Function for Adverse Pressure Gradient Flows
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.
A Crank-Nicolson Finite Difference Scheme for the (2+1)D Saturable Nonlinear Schrödinger Equation with Generalized Damping
In this study, we implement a Crank-Nicolson finite difference scheme to discretize the (2+1)D saturable nonlinear Schrödinger 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 Schrödinger equation without damping. Numerical simulations are conducted to validate these convergence properties and the conservation laws.
The Second-Order Mass-Conservative Characteristic Finite Difference Method for Sobolev Equation with Convection Term
In this work, a new Mass-Conservative Characteristic Finite Difference Method (MCC-FDM) is developed for solving the Sobolev equation. The new method combines the method of characteristics with mass-preserving interpolation, which not only keeps mass balance but also achieves second-order accuracy in both time and space. Numerical results are presented to confirm the temporal and spatial accuracy.
A Discrete Collocation Technique Using the Thin Plate Splines for Solving a Certain Class of Integro-Differential Equations Arising in Biology Models
The primary motivation of the current article is to study an approximate algorithm to solve a type of Volterra integro-differential equations involving nonlinear terms. These integro-differential equations have been utilized to simulate several realistic modeling arising in biological sciences, for example, the growth model of the toxins’ cumulative effects on a population residing in a closed system. Thin plate splines (TPSs), known as a subclass of radial basis functions (RBFs) independent of the shape parameter, create an efficient scheme to interpolate a function. Due to this desirable property, we use the discrete collocation approach with the TPS basis for estimating the solution of the mentioned integro-differential equations. Furthermore, the Gauss-Legendre integration formula is utilized to calculate the integrals emerging in the scheme. Remarkably, the algorithm of the offered approach can be comfortably performed on a PC with typical specifications derived from the simplicity of using TPSs. The researchers also investigate the convergence of the presented scheme. Some test examples including Volterra integro-differential equations are considered to make sure the veracity of the proposed scheme and the validity of the theoretical error analysis.
A Ciarlet-Raviart Mixed Finite Element Method for Optimal Control Problems Governed by a Fourth Order Bi-Wave Equation
The optimal control problem governed by a stationary fourth-order bi-wave equation is considered. To tackle this problem, we propose a bilinear mixed method of the Ciarlet-Raviart type. Our method exhibits an optimal convergence rate in the $L^2$-norm and demonstrates a global supercloseness property in the $H^1$-seminorm. Moreover, through the application of an interpolation post-processing technique, the method achieves global superconvergence. We provide two numerical examples to numerically validate these theoretical properties of our proposed method.
On Stable Difference Schemes for the Solution of Soliton Type Coupled Sine-Gordon Equations in the Weak Sense
In the present paper numerical solution of the nonlinear coupled system of sine-Gordon equations is studied in the weak sense. A first-order accurate and two second-order accurate unconditionally stable difference schemes corresponding to the system of sine-Gordon equations are considered. Solutions of these difference schemes are presented in the space of distributions by variational methods. The fixed point theory and the finite difference method are combined in the numerical implementations, carried out in MATLAB, in order to verify the theoretical results.
On Convergence and Superconvergence of Discontinuous Galerkin Method for Semi-Explicit Index-1 Integro-Differential Algebraic Equations
This paper mainly focuses on the discontinuous Galerkin (DG) method for solving the semi-explicit index-1 integro-differential algebraic equation (IDAE), which is a coupled system of Volterra integro-differential equations (VIDEs) and second-kind Volterra integral equations (VIEs). The DG approach is applied to both the VIDE and VIE components of the system. The global convergence respectively in the $L^2$-norm and $L^\infty$-norm is established, and the local superconvergence for VIDE component is obtained. Furthermore, numerical examples are presented to validate the theoretical convergence and superconvergence results.
Integro-differential algebraic equations
Copyright (c) 2025 COPYRIGHT: © Global Science Press
ALM-PINN: An Adaptive Physical Informed Neural Network Optimized by Levenberg-Marquardt for Efficient Solution of Singular Perturbation Problems
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.