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A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials. [ABSTRACT FROM AUTHOR]

In recent years, the heat kernel has proven to be a valuable tool for graph characterization and graph-based object clustering. It serves as the fundamental solution to the heat diffusion equation associated with the discrete graph Laplacian, describing how information flows across graph edges over time by exponentiating the Laplacian eigensystem. This paper focuses on the novel concept of the generalized heat kernel of a network, recently introduced by Kalala Mutombo et al. These authors build upon the k-path Laplacian operator for graphs, pioneered by Estrada et al., to incorporate long-range interactions (LRI) into information transmission across nodes and edges of a network/graph. LRI are handled through the use of Mellin and Laplace transforms. This paper makes a notable contribution by highlighting the practical application of the generalized heat kernel. We achieve this by demonstrating the effectiveness of the generalised heat kernel invariants in object clustering with real data using principal component analysis (PCA). Moreover, we investigate how incorporating long-range interactions (LRI) impacts object characterization and clustering, revealing superior results compared to conventional diffusion methods. Through experimentation, we show that object clustering remains achievable even with small values of the Mellin and Laplace parameters, contrasting with the requirement of an infinite value in the absence of LRI. • Application of graph generalised heat kernel invariants in object clustering via PCA. • Long-range interactions enhance graph-based object clustering. • Object clustering achieved with small Mellin/Laplace parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope (the 4-simplex) and tetrahedral prism elements. More generally, our objective is to construct high-order finite element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex in four dimensions. This paper is a natural extension of the companion paper entitled "Conforming finite element function spaces in four dimensions, part I: Foundational principles and the tesseract" by Nigam and Williams (2024). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers. [ABSTRACT FROM AUTHOR]

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed 'Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper the construction of C k basis functions on meshes with hanging nodes for hp -finite element methods is proposed. The construction is limited to axis-parallel, rectangular meshes, which can, however, be of any dimension. The basis functions are composed by using hierarchical tensor product shape functions which are defined on patches of so-called constraining nodes and combine univariate Hermite interpolation polynomials and Gegenbauer polynomials. The paper discusses how to integrate the constructed C k basis functions into the assembling process of stiffness matrices and load vectors by using connectivity matrices. Numerical examples illustrate the applicability of the proposed concept. [ABSTRACT FROM AUTHOR]

This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an Eulerian type of numerical finite difference approximation on structured grids for an advection dominated level set equation enhanced with a small curvature term. The proposed numerical scheme is based on upwind discretization in both the advection and the curvature term. This class of approximations is used in algebraic solvers, such as fast sweeping or fast marching methods, which respect the causality of solutions for which any information is propagating along characteristics, in order to solve the algebraic system in a finite number of computationally simple iterations or updates. An implicit numerical scheme for a time relaxed level set equation with non-zero right-hand side is used in this paper and solved by a nonlinear fast sweeping method. The results of the von Neumann stability analysis of the proposed scheme are presented, and numerical examples are used to verify the analysis. [ABSTRACT FROM AUTHOR]

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability. [ABSTRACT FROM AUTHOR]

Brain stroke is one of the most common causes of disability and death worldwide. In this paper, we present the numerical algorithm behind a new system for brain stroke detection: it relies on a novel imaging technique known as Electromagnetic Tomography (EMT). As compared to classical techniques, EMT has several advantages: it is noninvasive and, since it relies on nonionizing radiation, it is safe for the patient; it can be made compact and portable; it is easy to use and it has relatively low cost. The main drawback of EMT is its relatively poor spatial resolution due to restriction on the frequency spectrum. This paper describes the numerical challenges we had to overcome when designing our latest generation scanner. [ABSTRACT FROM AUTHOR]

The advection-diffusion equation is fundamental to modeling various transport phenomena, including the distribution of chemical species in surface or groundwater flow. In cases where the concentration at the source is unknown, inverse problem formulations are required to estimate the desired states by assimilating concentration monitoring data from specific points along the watercourse using a Bayesian approach. This paper proposes a combination of the Eulerian-Lagrangian method and the meshless method of fundamental solutions to solve the advection-diffusion equation. Moreover, the method was used as an evolution model for the sequential importance resampling particle filter algorithm to reconstruct the time-dependent inlet pollutant concentration in inverse problems. Numerical smooth and discontinuous inlet function results show that the particle filter - Eulerian-Lagrangian method of fundamental solutions combination can reconstruct inlet concentration time series. • Verification and validation of ELMFS solution with analytical, FDM and experimental data. • The ELMFS was coupled with sampling importance resampling particle filter for the solution of inverse problem. • The proposed framework is reliable and robust in reconstructing the inlet boundary conditions. [ABSTRACT FROM AUTHOR]

The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space. [ABSTRACT FROM AUTHOR]

The classic Landweber iteration method is widely used to solve ill-posed problems, and is proven to be very efficient. However the classic Landweber iteration method has the drawback of over smoothness near the cusps or jumps. Nowadays fractional regularization method becomes popular to overcome over-smoothness. In this paper we propose a fractional Landweber iteration method to solve a two-dimensional time-fractional inverse heat conduction problem. We give the selection of iterations under both a priori and a posteriori rules, and give the proof of error estimates between the exact solution and its regularization approximation. Further more, numerical results are included to verify the effectiveness of the proposed method. In experiment, we first calculate the direct problem of the heat conduction problem using the finite difference method; then we add random noise to the data and get the disturbed data; at last we compute the numerical approximate solution of the inverse problem with disturbed data by the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

Multiplicative noise removal is a challenging image processing problem. In this paper, we study a fractional-order nonlinear cross-diffusion system to denoise images corrupted by multiplicative Gamma noise. The system is coupled by two nonlinear, partial differential equations and gray level information, fractional derivative operator is considered in diffusion process to remove high level noise effectively while preserve structure details for texture images and SAR images. We prove the existence of weak solutions and other theoretical properties of the fractional-order cross-diffusion system under some hypotheses. In the numerical aspect, the model is implemented by virtue of the discrete fractional Fourier transform in the frequency domain. Experimental results show that the new system has a good compromise between multiplicative noise removal and structure preservation when compared to other well-known denoising methods. [ABSTRACT FROM AUTHOR]

As we know that, the parameterized Uzawa (PU) method can be very efficient when used to solve the standard saddle point problem, especially, when we have good and accurate estimation of preconditioned Schur complement matrix. In this paper, by taking full use of the special structure of coefficient matrix arising from complex Helmholtz equations, two types of optimized PU (OPU) methods are discussed theoretically and experimentally. Specifically, the convergence factors of these two OPU methods are less than 0.172, and the optimal result of first OPU method can reach 0.0396 with σ 1 ≥ σ 2 > 0 , which is currently the best theoretical result in the literatures. Moreover, the second OPU method has better computational advantages compared with the first OPU method as it avoids the inverse computation of W at each step, indicating the less CPU time will be costed by the second OPU method. In addition, the optimal parameters involved in our algorithms consist of constants, reducing the computational complexity associated with parameter selection. Finally, numerical results are given, not only show the effectiveness of OPU methods, but also confirm the rationality of theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the scale or shape parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the modified Navier-Stokes-Cahn-Hilliard equations with a data assimilation term to utilize the information of the observed data. Based on the idea of feedback control, this term nudges the solutions to the observed data sampled from the reference process. By utilizing the Crank-Nicolson formula and the scalar auxiliary variable approach, we introduce an efficient numerical scheme for the modified Navier-Stokes-Cahn-Hilliard equations. Properties of the mass conservation and unconditional energy stability are proved. We explore the robustness and efficiency of the proposed scheme with various experiments. [ABSTRACT FROM AUTHOR]

In this paper, we present a new model as an alternative to the classical 3D Richards model for the description of water flow in shallow aquifers. The new model is designed to achieve three goals. First, it provides a good approximation to the Richards model over a wide range of time scales. More specifically we show that both models characterize a flow with the same dominant components when the ratio of the horizontal length to the depth of the aquifer is small. Second, the new model accurately describes the velocity field. In particular, it is not based on the Dupuit hypothesis, which is often used in the context of shallow aquifers. This allows the new model to be well accurate even in the presence of wells and in aquifers with variable bedrock. Third, the new model can be viewed as a coupling of numerous 1D vertical Richards problems with a 2D elliptic one. In practice, this coupling is treated numerically using a Picard fixed point scheme, avoiding the need to solve any three-dimensional problem. This results in a significant reduction in computational cost compared to solving the 3D Richards problem directly. [ABSTRACT FROM AUTHOR]

In this paper, we devise a stochastic extragradient-like algorithm incorporated with inertial terms, which requires a single projection onto our feasible set and employs a stochastic approximation version of an Armijo-type line search scheme along a feasible direction, for solving pseudomonotone stochastic variational inequalities. In the algorithm, two different stepsize strategies are employed to update steplength sequences without using the prior knowledge of the Lipschitz constant of involved operators. In the process of the stochastic approximation, we iteratively reduce the variance of stochastic errors. The almost sure convergence, the complexity analysis, and rates are provided in a dimensional space under reasonable conditions. Finally, some numerical experiments with graphical illustrations are reported to demonstrate the applicability and the efficiency of our algorithm in comparison with some projection type methods in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an efficient numerical scheme to deal with heat transfer problems in pipe flow with a large length/diameter ratio. The generalized finite difference method (GFDM) is combined with a space stepping algorithm (SSA) for the solution. The SSA divides the solution domain into a number of overlapping sections in the length direction of the pipe. Due to the large Peclet number, the heat transport process is dominated by advection, which allows an approximate boundary condition to be applied at the downstream cross section. The problem is then solved section by section. Using the uniform distributed points, the solution matrix for each section does not change, and only the right hand side needs to be refreshed for the changing boundary conditions. This leads to a highly efficient scheme for problems with a large pipe length. To show the accuracy of the numerical scheme, a problem with available analytical solution is studied. Then the method is applied to problems of single pipe with different pipe wall temperature and flow Reynolds number, and concentric annular pipe. Numerical results confirm the stability and efficiency of the GFDM-SSA. The method can be applied to many real world transient heat transfer problems in which the pipe is heated or cooled along its length with arbitrary wall temperature for heat exchange and other purposes. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

In this paper we use the weighted discontinuous Galerkin finite element method to solve the dual-porosity-Brinkman coupling model. The dual-porosity model describes fluid flow in a dual-porosity medium composed of matrix and microfractures, while the Brinkman equation describes fluid flow in hydraulic fractures or conduits containing "debris filling" or "proppant filling". The two models are coupled by the four physically effective interface conditions, which are constructed based on the fundamental properties of the dual-porosity model and the Brinkman-Darcy model. We provide a weighted discontinuous discrete scheme for the model and derive the optimal error estimates. Through numerical experiments, we verify the properties of the model and the advantages of the numerical method, such as the optimal convergence rate of the numerical solution and the simulation of the flow characteristics around hydraulic fractures and conduits. [ABSTRACT FROM AUTHOR]

In this paper, staggered discontinuous Galerkin (SDG) approximation for the Stokes problem is developed in a 2D square domain. The square domain is discretized by using rectangular meshes, which allow us easy identification of elements and hence a simple implementation of the SDG methods. By introducing proper function spaces and interpolations for the velocity vector field, its gradient, and the pressure, some required inf-sup conditions of our SDG method are shown. Then, thanks to the inf-sup conditions, the optimal convergence of our SDG approximation to the Stokes solution is obtained with respect to the polynomial order. Numerical results are included as well to validate the performance of our proposed SDG method for the Stokes problem. [ABSTRACT FROM AUTHOR]

This paper investigates the stabilization of Euler-Bernoulli beam systems modeled by a partial differential equation (PDE) with unknown time-varying disturbance. An iterative learning controller is designed using only boundary state feedback to realize the vibration control subject to unknown boundary disturbance. The well-posedness for the closed-loop system is given by the operator semigroup theory. Furthermore, the exponentially stable for the closed-loop system is proved by the Lyapunov method. The comparisons with existing results are made to demonstrate the effectiveness and advantages of the proposed boundary iterative learning control method. [ABSTRACT FROM AUTHOR]

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

Numerical studies of flow in square enclosures attain great interest due to their wide applications in engineering, such as in the meteorology, navigation, mechanology and mining, etc. In this paper, the solutions are obtained to the square lid driven cavity describing classical mixed convection in the presence of an inclined magnetic field, chemical radiation and heat absorption/generation. The problem is mathematically reduced into Navier-Stokes equations. The Coiflet wavelet homotopy approach, a newly suggested computational technique, is then used to solve these nonlinear partial differential equations with their non-homogeneous boundary conditions. The numerical solutions for streamlines, isotherms, iso-concentrations contours, Nusselt number are carefully determined. The solutions so obtained are analyzed through various plots to demonstrate the effects of several physical parameters such as Reynolds numbers (Re), Richardson number (Ri), Grashof number (Gr), Hartman number (Ha), thermal radiation parameter (N R), chemical reaction coefficient (K), heat generation/absorption coefficient (Q h), the angle of inclination of magnetic field (γ), slipping parameter (λ) and Schmidt number (Sc) and their physical insights are also reported. Rigid comparison of with previous studies shows excellent computational effectiveness of newly proposed method. Additionally, according to the results, our recently proposed approach has superior characteristics and strong nonlinear processing capabilities, which render it more appropriate than the Coiflet wavelet analysis method and the traditional homotopy analysis method for tackling complex nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format. [ABSTRACT FROM AUTHOR]

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. [ABSTRACT FROM AUTHOR]

This paper analyzes a parareal approach based on fractional-step methods for the nonstationary Navier-Stokes equations. As an efficient parallel computing framework, the coarse propagator often determines the performance of the parareal algorithm. We present a parareal algorithm using the fractional-step method, a very efficient time discrete scheme for the Naiver-Stokes equations, as the coarse propagator for the Navier-Stokes equations. Then we give the specific stability and convergence analysis of this specific parareal algorithm. Finally, numerical experiments are done to show efficiency and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]

In Beirão da Veiga et al. 2019, the authors proposed a virtual element method for domains with fixed curved boundary that tends to be locally straight when the discretisation gets finer and finer. In this paper, we properly modify this method so that it can also deal with curved mesh edges which are not necessarily located along boundaries and do not tend to be straight when refining the mesh. To achieve this goal, we assume that curved edges are described by polynomials and we increase the dimension of the virtual element space used in Beirão da Veiga et al. 2019. In the numerical experiments, we compare these two methods. Furthermore, we apply the proposed approach to the benchmark "TEAM 25" problem, an optimal shape design problem in magnetostatics characterised by curved edges. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the convergence rates of an implicit-explicit (IMEX) difference scheme for solving a two-dimensional partial integro-differential equation (PIDE) with an inexact boundary which arises in option pricing with stochastic intensity jumps. First, the IMEX scheme is proposed to solve the two-dimensional PIDE and its inexact boundary governed by a one-dimensional PIDE. Then the second-order convergence rates of the IMEX scheme for the main PIDE are proved for both time and space based on the second-order convergence analysis in the discrete H 1 -norm of the IMEX scheme for the boundary PIDE. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, according to the element discretization characteristic of the geometric model, a complex function theory based transformation boundary element method is proposed to perform the steady-state thermal analysis of plate structures. The numerical implementation of the proposed method can be attributed to a numerical integration problem in element analysis, which is mainly divided into the following two steps: firstly, a bidirectional transformation method based on distance function and complex function is proposed to remove the weak singularities of the fundamental solution in integral equation, which is suitable for the weakly singular integrals of narrow strip discrete elements; secondly, a dual sinh method based on complex poles is developed to remove the influence of the nearly singular integrals in the analysis of the plate structures. With this method, the numerical integration in the integral equation can be accurately evaluated, which ensures the accurate calculation of the thermal physical variables of the plate structures. Several numerical examples are presented and discussed to prove the accuracy and versatility of the presented method. The results show that the presented method has the superiority of dealing with plate structure problems. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating a second-order time parallel decoupled algorithm for the mixed Stokes/Darcy model. The main objective of this algorithm is to deal with the issue of low computational efficiency in solving the coupling problem directly. Our scheme not only reduces the computation scale through the decoupling method, but also realizes the parallelism in the time direction through parareal method. Furthermore, we implement a second-order spectral deferred correction method to improve accuracy. For some small time step size, we prove the stability and convergence results of the time parallel decoupled algorithm. Finally, we provide three numerical tests to demonstrate the reliability and computational efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An H (d i v) -conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the L a m e ´ parameter λ in error estimates, i.e., making the scheme locking-free. The method works without requiring λ ‖ ∇ ⋅ u ‖ 1 to be bounded. We prove optimal error estimates, independent of λ , in both the H 1 -norm and the L 2 -norm. Numerical experiments validate that the method is effective and locking-free. [ABSTRACT FROM AUTHOR]

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves, relaxation oscillations and spatio-temporal Turing instability. Inspired by the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of certain reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, a λ - ω system and the DIB morpho-chemical system for battery modeling. Finally, a discussion about the overall computational load and the future prediction features of the new algorithm is also provided. [ABSTRACT FROM AUTHOR]

In this paper, an isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed for the time-domain acoustic wave problem in 3D infinite domain. The fundamental solution of the potential problem is used to establish the boundary-domain integral equation, which avoids the problem of solving the coefficient matrix repeatedly at different times. On the one hand, in order to maintain the dimensionality reduction advantages of the boundary element method, the classical dual reciprocity method is used to transform the domain integral into a boundary integral. On the other hand, for purpose of satisfying the boundary conditions at infinite distance accurately, a reliable integral convergence criterion is established by adopting the approximate variation coefficient. Furthermore, the effects of different approximation functions, variation coefficients, time steps, number of interior points and elements on the results are discussed in detail by several typical examples. Numerical results show that the proposed method has high accuracy and good stability even when analyzing geometrically complex dolphin models. • IGABEM for solving acoustic wave in time domain is proposed for the first time. • The coefficient matrix formed by the boundary integral equation only requires to be calculated once. • The adaptive expansion basis function satisfying the integral regularization condition is proposed. • The multi-patch dolphin model is solved and a stable solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, a simple preconditioner with two parameters (SPTP) is introduced for solving a class of saddle-point problems. The SPTP preconditioner is block upper triangular and does not contain the approximate matrix of the Schur complement matrix. Theoretical analyses on the sufficient conditions under which the SPTP iterative sequence converges to the unique solution, are given in detail. Convergence conditions in some special cases are also analyzed. Moreover, detailed analyses on the theoretical optimal parameters and corresponding optimal convergence factor of the SPTP method are given. Some numerical examples show that the SPTP iterative method outperforms several SOR-type methods by using theoretical optimal parameters simultaneously. Some newly proposed methods are also compared with the SPTP iterative method by using the experimental optimal parameters simultaneously, and the SPTP iterative method is still advantageous, from the point of view of iteration step and CPU time. Moreover, we also investigate relatively practical values of the parameters since searching for the optimal parameters will waste a lot of time in practice. [ABSTRACT FROM AUTHOR]

The Volume of Fluid (VoF) method stands out as a widely utilized tool for capturing interfaces in the numerical simulation of multimaterial fluid flow. Numerous efforts have been invested in enhancing its accuracy and efficiency, including the development of analytic reconstruction methods. Despite these advancements, there exists a continued need for further improvements in the accuracy, efficiency, and robustness of the VoF method in practical applications. In the conventional VoF method, information from neighboring cells is crucial for computing the normal vector of the interface in each multimaterial cell, requiring data from as many as 26 neighboring cells in 3D VoF methods. This paper introduces a novel VoF method named SVoF, which departs from traditional approaches by eliminating the necessity for information from neighboring cells. In the SVoF method, the data provided for interface reconstruction in each multimaterial cell consists of the volume fractions of fluids in several subcells within the cell itself. Initially, these data are determined by the initial conditions. In MMALE methods, these data can be updated through a closure model for multimaterial cells in each Lagrangian stage and remapped in each subsequent remapping stage. The determination of the normal vector of the linear interface in SVoF is achieved through an optimization procedure. The objective function of this procedure is the sum of the squares of the differences between the given volume fractions and the reconstructed volume fractions of the reference fluid in specific subcells. The optimization procedure is solved using an improved quadratic fit golden section algorithm. Similar to traditional VoF methods, the intercept of the linear interface is calculated by precisely matching the given volume fraction of the reference fluid throughout the entire multimaterial cell. The SVoF method is characterized by its ease of coding and implementation, making it suitable for interface reconstruction in multimaterial cells containing three materials. Building upon the SVoF method, a new cell-centered MMALE method is developed, showcasing robustness. This new MMALE method utilizes Tipton's pressure relaxation model for the closure of multimaterial cells and leverages the SVoF method for interface reconstruction. Numerical results demonstrate that the SVoF method can precisely reproduce linear interfaces, approaching second-order accuracy, and that this new MMALE method exhibits significant robustness. • A novel VoF method (SVoF) that doesn't require information from neighboring cells. • SVoF requires the volume fractions of fluids in multiple subcells within each multimaterial cell. • SVoF is applicable for interface reconstruction in multimaterial cells containing three distinct materials. • SVoF precisely reproduces linear interfaces and is straightforward to code and implement. • MMALE combined with SVoF demonstrates greater robustness compared to the existing MMALE coupled with MoF. [ABSTRACT FROM AUTHOR]

Research of numerical methods for solving Maxwell's equations in Kerr-type nonlinear media is quite popular. A so-called leapfrog alternating direction implicit (ADI) method which avoids mid-time computations is both unconditionally stable and computationally efficient, but hard to be applied to nonlinear problem. In this paper, we based on the difference between leapfrog method and leapfrog ADI method of linear Maxwell's equations, developed a modified leapfrog (ML) method for Maxwell's equations in Kerr-type nonlinear media. Stability and error estimate of the ML method are discussed. Numerical results have been achieved to verify the unconditional stability, and second-order convergence rate in both time and space, and ML method is more efficient than Crank-Nicolson (CN) method. [ABSTRACT FROM AUTHOR]

This paper proposes a novel matrix assembly method called dummy scalar triplet to construct the implicit lattice Boltzmann flux solver (LBFS). The implicit discretize scheme enables the LBFS to use a time step far exceeding the Courant-Friedrichs-Lewy condition limit. Therefore, the computational efficiency and the applicability of LBFS are significantly enhanced, especially in solving complex geometry problems. In addition, a projection-based interpolation scheme is introduced to calculate the flux across the cell interface. Compared with the conventional interpolation scheme, the projection-based scheme has a higher accuracy, and its stability is hardly affected by the streaming time step. Three simulations on isotropic and anisotropic convection-diffusion equations are conducted to validate the improved method, and all the results obtained are in good agreement with the analytical solutions. • Propose an implicit construction approach for the lattice Boltzmann flux solver. • The construction approach can be easily applied to different governing equations. • Allow using time steps far beyond the Courant-Friedrichs-Lewy condition limit. • The conventional interpolation method may generate dummy flux at cell interfaces. • An accurate and stable projection-based interpolation is proposed. [ABSTRACT FROM AUTHOR]

In this paper, a new hybrid numerical approach is presented that couples "ghost cell" based immersed boundary (IB) method with the finite volume lattice Boltzmann method (FVLBM). In the implementation process, the grid cells are classified into three types, i.e., " f l u i d c e l l ", " s o l i d c e l l " and " g h o s t c e l l ", where the " g h o s t c e l l " is the first layer of " s o l i d c e l l " near the " f l u i d c e l l ". As the wall boundary condition is reflected in the " g h o s t c e l l ", the high–accuracy reconstruction of variables in it will be the key point of present approach. Then, a FVLBM scheme is used to update the distribution function in the " f l u i d c e l l ", and a least square interpolation based formulation is derived to construct the distribution functions in it. Further, the bounce–back scheme embedded in the streaming–collision lattice Boltzmann method will be employed as the wall boundary condition. Besides, for the class of moving boundary problems, the construction of the distribution function in the " f r e s h c e l l " is also considered. A number of typical benchmarking incompressible viscous fluid flows over both stationary and moving objects are simulated to justify the present method, using obstacles ranging from a stationary and a translating circular cylinders, a rotationally oscillating cylinder, to two more complex flows around an oscillating cylinder in the stationary fluid and in the free–stream. Besides, the temporal and spatial accuracy also are testified by the simulation of Taylor–Couette flow. The obtained accurate simulation results demonstrate the capability of the present hybrid IB–FVLBM for computer simulations of both stationary and moving boundary problems in the incompressible flow regime. [ABSTRACT FROM AUTHOR]

This paper presents a study of large linear systems resulting from the regular B -splines finite element discretization of the c u r l − c u r l and g r a d − d i v elliptic problems on unit square/cube domains. We consider systems subject to both homogeneous essential and natural boundary conditions. Our objective is to develop a preconditioning strategy that is optimal and robust, based on the Auxiliary Space Preconditioning method proposed by Hiptmair et al. [48]. Our approach is demonstrated to be robust with respect to mesh size, and we also show how it can be combined with the Generalized Locally Toeplitz (GLT) sequences analysis presented in [59] in order to derive an algorithm that is optimal and stable with respect to spline degree. Numerical tests are conducted to illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

This paper proposes a Cahn-Hilliard type inpainting model equipped with a nonlocal diffusion operator. A rigorous analysis of the well-posedness of the stationary solution is established using Schauder's fixed point theory. We construct a time stepping scheme based on the convex splitting method with the nonlocal term treated implicitly and the fidelity term treated explicitly. We prove the consistency, stability and convergence of the semidiscrete-in-time scheme. To the best of our knowledge, this is the first study to present such an analysis for semidiscrete-in-time problems of this model, which provides valuable guidance for parameter selection. Numerical experiments validate the effectiveness of the proposed nonlocal model, which shows superior performance compared to both local and classical total variation models in preserving fine textures and recovering image edges. [ABSTRACT FROM AUTHOR]

Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is physics-informed neural networks (PINN), which integrate physical principles into neural networks. This transforms the process of solving PDEs into optimization problems for neural networks. To address a collection of advection-diffusion equations (ADE) in a range of difficult circumstances, this paper proposes a novel network structure. This architecture integrates the solver, a multi-scale deep neural networks (MscaleDNN) utilized in the PINN method, with a hard constraint technique known as HCPINN. This method introduces a revised formulation of the desired solution for ADE by utilizing a loss function that incorporates the residuals of the governing equation and penalizes any deviations from the specified boundary and initial constraints. By surpassing the boundary constraints automatically, this method improves the accuracy and efficiency of the PINN technique. To address the "spectral bias" phenomenon in neural networks, a subnetwork structure of MscaleDNN and a Fourier-induced activation function are incorporated into the HCPINN, resulting in a hybrid approach called SFHCPINN. The effectiveness of SFHCPINN is demonstrated through various numerical experiments involving ADE in different dimensions. The numerical results indicate that SFHCPINN outperforms both standard PINN and its subnetwork version with Fourier feature embedding. It achieves remarkable accuracy and efficiency while effectively handling complex boundary conditions and high-frequency scenarios in ADE. [ABSTRACT FROM AUTHOR]

This paper investigates the convergence behavior of the Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations. The algorithms are applied to regular domains in 1D and 2D for multiple subdomains, and the impact of different constant values of the generalized diffusion coefficient on the algorithms' convergence is analyzed. The convergence rate of the algorithms is analyzed as the fractional order of the time derivative changes. The paper demonstrates that the algorithms exhibit slow superlinear convergence when the fractional order is close to zero, almost finite step convergence (exact finite step convergence for wave case) when the order approaches two, and faster superlinear convergence as the fractional order increases in between. The transitional nature of the algorithms' behavior is effectively captured through estimates with changes in the fractional order, and the results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]