We propose a level set method for a Stokes eigenvalue optimization problem. A relaxed approach is employed first to approximate the Stokes eigenvalue problem and transform the original shape optimization problem into a topology optimization model. Then the distributed shape gradient is used in numerical algorithms based on a level set method. Single-grid and efficient two-grid level set algorithms

Fast and accurate predictions of the spatiotemporal distributions of temperature are crucial to the multi-scale thermal management and safe operation of microelectronic devices. To realize it, an efficient semi-implicit Lax-Wendroff kinetic scheme is developed for numerically solving the transient phonon Boltzmann transport equation (BTE) from the ballistic to diffusive regime. The biggest innovation

In order to provide a more effective method to describe the local structure of the degraded image and to enhance the robustness of the denoising, we propose a non-convex total variational image denoising model that combines the non-convex log function with an adaptive weighted matrix within the total variation framework. In the proposed model, the weighted matrix is capable of effectively describing

This study employs an integral transform approach for Love wave propagation in a rotating composite structure having an interfacial crack. The structure comprises an initially stressed functionally graded piezoelectric-viscoelastic half-space bonded to a piezoelectric-viscoelastic half-space, and is subjected to anti-plane mechanical loading and in-plane electrical loading. The study focuses on two

In this article, a mixed spectral element method combined with second-order time stepping schemes for solving a two-dimensional nonlinear fourth-order fractional diffusion equation is constructed. For formulating an efficient numerical scheme, an auxiliary function is introduced to transform the fourth-order fractional system into a low-order coupled system, then the time direction is discretized by

In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure pairs are residual-based stabilisations. When using first-order elements, however, the viscous part of the residual cannot be approximated, which often degrades accuracy. For constant viscosity, or by assuming a Lipschitz condition on the viscosity field, we can construct stabilisation

The two-dimensional (2D) Kuramoto-Sivashinsky equation offers a robust framework for studying complex, chaotic, and nonlinear dynamics in various mathematical and physical contexts. Analyzing this model also provides insights into higher-dimensional spatio-temporal chaotic systems that are relevant to many fields. This article aims to solve the scalar form of the two-dimensional Kuramoto-Sivashinsky

This study investigates the application of wetting boundary conditions for modelling flows in complex curved geometries, such as rough fractures. It implements and analyses two common variants of the wetting boundary condition within the three-dimensional (3D) phase field lattice Boltzmann method. It provides a straightforward and novel extension of the geometrical approach to curved three-dimensional

Traditional transform thermodynamic devices are designed from anisotropic materials which are difficult to fabricate. In this paper, we design and simulate a carpet thermal concentrator. Based on existing transformation thermodynamic techniques, we have derived the perfect parameters required for carpet heat concentrators. In order to eliminate the anisotropy of perfect parameters, we designed a heat

The lattice Boltzmann method (LBM) for compressible flow is characterized by good numerical stability and low dissipation, while the conventional finite volume solvers have intrinsic conversation and flexibility in using unstructured meshes for complex geometries. This paper proposes a strategy to combine the advantages of the two kinds of solvers by designing a finite volume solver to mimic the LBM

We consider the coupled system of the Landau–Lifshitz–Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For this nonlinear system of time-dependent partial differential equations, we present a decoupled integrator based on first-order finite elements in space and an implicit

In this research article, we have simulated the solutions of three types of (classical) moving boundary models in ductal carcinoma in situ by an efficient temporal-spatial spectral collocation method. In all of these three classical models, the associated fixed (spatial) boundary equations are localized by the numerical scheme. In the numerical scheme, Laguerre polynomials and Hermite polynomials are

This paper presents a phase field approach to multi-material topology optimization of thermo-elastic structures. Based on the ordered Solid Isotropic Material with Penalization (ordered SIMP) model, the phase field variable is interpreted as the normalized density, which is used as the design variable in topology optimization. The material properties are interpolated in each interval of the normalized

We propose mixed finite element methods for the coupled Biot poroelasticity and Poisson–Nernst–Planck equations (modeling ion transport in deformable porous media). For the poroelasticity, we consider a primal-mixed, four-field formulation in terms of the solid displacement, the fluid pressure, the Darcy flux, and the total pressure. In turn, the Poisson–Nernst–Planck equations are formulated in terms

Based on the primal hybrid finite element method (FEM) to discretize spatial variables, a semi-discrete scheme is obtained for the weakly damped Klein-Gordon equation. It is shown that this method is energy-conservative, and optimal error estimates in the energy norm are proved with the help of a modified elliptic projection. Moreover, a superconvergence result is derived, and as a consequence, the

A fully discrete implicit scheme is presented and analyzed for the nonlinear Sobolev equations, which combines an anisotropic spatial nonconforming FEM with the variable-time-step BDF2 such that nonuniform meshes can be adopted in both time and space simultaneously. We prove that the fully discrete scheme is uniquely solvable, possesses the modified discrete energy dissipation law, and achieves second-order

In this paper, we propose a numerically stable and efficient method based on Haar wavelets for solving high-dimensional second-order parabolic partial differential equations (PDEs). In the proposed framework, the spatial second-order derivatives in the governing equation are approximated using the Haar wavelet series. These approximations are subsequently integrated to obtain the corresponding lower-order

A challenging and common problem that frequently arises in the fields of physics and engineering, two-dimensional (2D) incompressible, viscous MHD duct flows have significant theoretical and practical significance due to their numerous and widespread applications in astrophysics, geology, power generation, MHD generators, electromagnetic pumps, accelerators, blood flow measurements, drug delivery,

Variational Physics-Informed Neural Networks often suffer from poor convergence when using stochastic gradient-descent-based optimizers. By introducing a least squares solver for the weights of the last layer of the neural network, we improve the convergence of the loss during training in most practical scenarios. This work analyzes the computational cost of the resulting hybrid least-squares/gradient-descent

We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete variational problem. In the full space-time formulation, time is considered another dimension, and the time derivative is interpreted as an additional advection term

In this paper, we study an asymptotic preserving (AP), energy stable and positivity preserving semi-implicit finite volume scheme for the Euler-Poisson-Boltzmann (EPB) system in the quasineutral limit. The key to energy stability is the addition of appropriate stabilisation terms into the convective fluxes of mass and momenta, and the source term. The space-time fully-discrete scheme admits the positivity

This study develops a new family of explicit time integration methods for linear structural dynamic analysis. The proposed method is formulated using cubic B-spline interpolation. Several cases of algorithm parameters are identified by theoretical analysis to improve stability and accuracy. The explicit method exhibits desirable algorithmic properties, including stability and accuracy. The numerical

This paper proposes a physics-informed radial basis function network (RBFN) based on Hausdorff fractal distance to resolve Hausdorff derivative elliptic problems. In the proposed scheme, we improve the performance of RBFN via setting the source points outside the computational domain, and allocating distinct shape parameter values to each RBF. Furthermore, on the basis of the modified RBFN, we take

In the current study, we propose an accurate numerical method for plane elastostatic equations of anisotropic functionally graded materials. The proposed method uses radial basis functions augmented with polynomial basis functions in a collocation framework by employing ghost point centers which cover physical domain of considered problem. Unlike in classical collocation approach where the centers

The laser-induced thermal therapy (LITT) scheme has proved great efficacy in tumor treatment. Therefore, the research between the heat conduction problems of LITT has become a hot topic in recent years. To seek rational constitutive relations of heat flux and temperature which can describe the heat transfer behavior of LITT, we develop a novel distributed-order time fractional derivative model based

We propose a scalable well-balanced numerical method to efficiently solve a modified set of shallow water equations targeting the dynamics of lava flows. The governing equations are an extension of a depth-integrated model already available in the literature and proposed to model lava flows. Here, we consider the presence of vents that act as point sources in the mass and energy equations. Starting

The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps across mesh interfaces from the FEM stiffness bilinear form while maintaining the system's coercivity. Herein, we present two generalizations for SoftFEM that aim to

We develop a novel mixed method for addressing two-dimensional Laplacian problem with Dirichlet boundary conditions, which is recast as a rot-div system of three first-order equations. We have established the well-posedness of this new method and presented the a priori error estimates. The numerical applications of Bercovier-Engelman and Ruas test cases are developed, assessing the effectiveness of

In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calderón problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient

In this study, using implicit Euler method and second-order centered difference methods to approximate the first-order temporal and second-order spatial derivatives, respectively, introducing a stabilized term and applying u(xi,yj,tk)−[u(xi,yj,tk)]pu(xi,yj,tk+1) to approximate the nonlinear term u(xi,yj,tk+1)−[u(xi,yj,tk+1)]p+1 at (xi,yj,tk+1), a class of stabilized, non-negativity- and boundedness-preserving

In this paper we introduce and analyze an hp finite element method to solve a non-standard spectral problem in a curved plane domain using curved elements. This problem arises from nuclear engineering: the vibration of elastically mounted tubes immersed in a cavity filled with fluid. The eigenvalue problem is presented in a proper setting and we prove, under appropriate assumptions about the curved

This paper presents a novel physics-informed neural network (PINN) architecture that employs exponential basis functions (EBFs) to solve many boundary value problems. The EBFs are organized so that the PINN architecture may employ their simple differentiation features to solve partial differential equations (PDEs). The proposed approach has been meticulously investigated and compared to the conventional

In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special partial Least-Squares (Galerkin-least-squares) method, known as the augmented mixed finite element method, and its relationship to the standard least-squares finite element method (LSFEM). Two versions of augmented mixed finite element methods are proposed in the paper with robust

This paper concerns the sparse optimal control problem subject to the monodomain equations. Monodomain equations are coupled equations that model the electrophysiological wave propagation of the action potential in cardiac muscle. This model consists of a reaction-diffusion PDE coupled with an ODE. A non-smooth term is added to the cost in addition to the usual quadratic cost so that the optimal control

In this paper, we propose a nonlinear time-space fractional diffusion model to remove the multiplicative gamma noise. This model incorporates Caputo time-fractional derivative into the existing space-fractional diffusion models. It leverages the memory effect of time-fractional derivatives to control the diffusion process, achieving a balance between edge preservation, texture retention and denoising

The Motz problem of 1946 has attracted considerable interest for numerical schemes to accommodate the singularity due to a switch in boundary conditions from Dirichlet to Neumann at a mid-point on one side of a rectangular domain defined by Laplace's equation. Although a detailed solution was provided by means of conformal transformations in 1972 for a harmonic potential function, that solution is

This paper introduces meshless and high-precision barycentric interpolation collocation methods, grounded in two well-established difference formulas, for solving the nonlinear time-fractional Schrödinger equation. This equation is characterized by complexity arising from power-law nonlinearity and multi-scale time dependence. To enhance spatial accuracy, we utilize two meshless barycentric interpolation

The main goal of this paper is to extend the numerical scheme for the transport equation described in previous works [Penel, 2012; Bernard et al., 2014] from one to two dimensional problems. It is based on the method of characteristics, which consists in solving two ordinary differential equations rather than a partial differential equation. Our scheme uses an adaptive 6-point stencil in order to reach

In this paper, an ensemble-based efficient iterative method is used to solve the partial differential equations (PDEs) with random inputs. The aim of the efficient iterative method is to get a good approximation of the Galerkin solution for PDEs with random inputs. An essential ingredient of the proposed method is to construct the decomposition of stochastic functions, involving parameter-independent

The momentum exchange method (MEM) is widely used to calculate hydrodynamic forces on solid particles in the lattice Boltzmann method. Although MEM achieves second-order accurate force computation on particles with appropriate bounce-back schemes, significant numerical fluctuations can occur when particle moves relative to the mesh lines. In this work, we extend the recent analysis of Dong et al. [1]

The computation accuracy of non-conforming isoparametric elements in the displacement finite element method remains suboptimal when confronted with serious mesh distortion. To improve this issue, the area coordinate system and the volume coordinate system method based on displacement were proposed in the last century. By adopting volume coordinates as local coordinates and integrating the advantages

This study presents an in-depth analysis of the nonconforming virtual element method (VEM) as a novel approach for approximating the eigenvalues of the Schrödinger equation. Central to the strategy is deploying the L2 projection operator to discretize potential terms within the model problem. Through compact operator theory, we rigorously establish the methodology's capability to achieve double-order

We develop a peridynamic model, called non-local bond-based linear peridynamics model, for viscoelastic materials based on the fractional derivatives, which captures power-law responses and so provides a very competitive description of the mechanical vibrations. We accordingly derive a numerical scheme, efficient collocation method, to simulate the viscoelastic nonlocal model. Finally, we investigate

Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs), but there still remain some challenges in the application of PINNs, for example, how to exhaustively utilize a small size of (usually very few) labeled samples, which are the exact solutions to the PDEs or their high-accuracy approximations, to improve the accuracy

This paper establishes a numerical framework for addressing the inverse problem of PT-symmetric potentials. Firstly, we discretize the solution space and innovatively construct a mapping to project the inverse problem of the PT-symmetric potential onto a finite-dimensional real vector space, thereby transforming the inverse problem of PT-symmetric potentials in the complex domain into a root-finding

This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the H(div) conforming virtual element method for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration

This paper reports a class of new hybrid compact finite-difference schemes with high-order accuracy for solving the Helmholtz equation in two and three dimensions. The innovation of the scheme is to hybridize explicit and implicit compact schemes to deal with the solution and its first- and second-order derivatives, and to solve it by a step-by-step coupled iterative method. In response to the inefficiency

The virtual element method (VEM) was proposed for the nonlinear convection-diffusion-reaction problem in [5]. Using projection operators, a computable VEM discrete scheme was derived and the existence of the solution was proved. However, even when higher order elements were introduced, the SUPG framework shows spurious oscillations in the crosswind direction. In this paper, we propose, in the context

Fixed-point or Newton-methods are typically employed for the numerical solution of nonlinear systems arising from discretization of nonlinear magnetic field problems. We here discuss an alternative strategy which uses Quasi-Newton updates locally, at every material point, to construct appropriate linearizations of the material behavior during the nonlinear iteration. The resulting scheme shows similar

Image inpainting is a technique for reconstructing missing or damaged regions of an image. In this paper, we propose a novel linear numerical method with second-order accuracy in both space and time for solving the modified Allen–Cahn equation applied to image inpainting. The proposed method is conditionally maximum principle-preserving, second-order accurate, and unconditionally energy-stable. A leap-frog

We present a highly efficient solver, accelerated by graphics processing units (GPUs), for simulations of turbulent flows over wave surfaces. The solver employs a boundary-fitted curvilinear grid, which is horizontally discretized by a Fourier-based pseudospectral scheme, enabling accurate and efficient resolution of turbulence motions and wave geometry effects across scales. Our GPU implementation

This work presents novel extensions for combining two frameworks for quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainties in the modeling of engineered systems. The data-consistent (DC) framework poses an inverse problem and solution for quantifying aleatoric uncertainties in terms of pullback and push-forward measures for a given Quantity of Interest

This paper studies linear and unconditionally energy stable schemes for the functionalized Cahn-Hilliard (FCH) equation. Such schemes are built on the exponential scalar auxiliary variable (ESAV) approach and the one-parameter time discretizations as well as the extrapolation for the nonlinear term, and can arrive at second-order accuracy in time. It is shown that the derived schemes are uniquely solvable

In this paper, an efficient spectral-Galerkin method is proposed to solve the fourth order elliptic equation with a variable coefficient on the unit disc. The efficiency of the method is rest with the use of properly designed orthogonal polynomials as basis functions, which preserve the block-diagonal matrix structure of the discretized system with a constant coefficient and also work well with the

In this paper, we study the phase field model on a three-dimensional bounded domain for a two-phase, incompressible, inductionless magnetohydrodynamic (MHD) system, which is important for many engineering applications. To efficiently and accurately solve this multi-physics nonlinear system, we present a fully discrete scheme that ensures both mass and charge conservation. Making use of the discrete

This study introduces a first step for constructing a hybrid reduced-order models (ROMs) for segregated fluid-structure interaction in an Arbitrary Lagrangian-Eulerian (ALE) approach at a high Reynolds number using the Finite Volume Method (FVM). The ROM is driven by proper orthogonal decomposition (POD) with hybrid techniques that combines the classical Galerkin projection and two data-driven methods

This paper proposes a staggered discontinuous Galerkin (SDG) method for solving the 2D SN transport equation on arbitrary polygonal mesh. The new method allows rough grids such as highly distorted quadrilateral grids and general polygonal grids. More importantly, it is numerical flux free, different from the standard discontinuous Galerkin (DG) method using upwind flux, and thus gains the advantage

In this study, a novel class of block ω-circulant preconditioners is developed for the all-at-once linear system that emerges from solving parabolic equations using first and second order discretization schemes for time. We establish a unifying preconditioning framework for ω-circulant preconditioners, extending and modifying the preconditioning approach recently proposed in (Zhang and Xu, 2024 [27])