An improved boundary knot method (IBKM) is proposed to enhance the performance of BKM in solving homogeneous high-order Helmholtz-type partial differential equations. Compared with the classical BKM where the sources are always placed on the physical boundary as collocation points, the new sources named fictitious points are now placed on multi-layer extended pseudo boundaries. This modification leads

For the fourth order time-dependent singularly perturbed Bi-wave equation modeling d-wave superconductors, the implicit Backward Euler (BE) and Crank-Nicolson (CN) schemes of Galerkin finite element method (FEM) are studied by Bonner-Fox-Shmite element. Then the quasi-uniform and unconditional superconvergent error estimates of orders O(h3+τ) and O(h3+τ2) (h, the spatial parameter, and τ, the time

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under

This paper primarily focuses on developing a high-order-in-time spectral Galerkin approximation method for nonlinear stochastic Fisher's type equations driven by multiplicative noise. For this reason, we first design an improved discretization scheme in time based on the Milstein method, and then propose a spectral Galerkin approximation method in space. We analyze the H1 stability and L2 stability

This article aims at studying new finite difference methods for one-dimensional and two-dimensional nonlinear Riesz space variable-order (VO) fractional diffusion equations. In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Grünwald-difference (4WSGD) operators for Riemann-Liouville constant-order fractional derivatives, which

In this paper, we consider higher order multipoint flux mixed finite element methods for parabolic problems. The methods are based on enhanced Raviart-Thomas spaces with bubbles. The tensor-product Gauss-Lobatto quadrature rule is employed, which enables local velocity elimination and results in a symmetric, positive definite cell-based system for pressures. We construct two fully discrete schemes

In this paper, we present and analyze a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme for the FitzHugh-Nagumo equation. Utilizing a second-order finite-difference space discretization, we derive the fully discrete numerical scheme by incorporating both the stabilization technique and the ETDRK2 scheme for temporal approximation. The smallest invariant set of the FitzHugh-Nagumo

In the analysis of transient seepage flow with free surfaces, not only the free surfaces but also the boundary conditions vary with time, introducing significant challenges to those traditional mesh-based numerical methods. Although the numerical manifold method (NMM) has shown great advantages in tracking time-independent free surface seepage flow due to its dual cover systems – the mathematical cover

Chew, Goldberger & Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem

This paper investigates the uniform convergence of arbitrary order finite element methods on Vulanović-Bakhvalov mesh. We carefully design a new interpolation based on exponential layer structure, which not only overcomes the difficulties caused by the mesh step width, but also ensures the Dirichlet boundary condition. We successfully demonstrate the uniform convergence of the optimal order in the

The objective of this paper is to present efficient numerical algorithms to resolve the two-dimensional fractional Oldroyd-B model. Firstly, two compact alternating direction implicit (ADI) methods are constructed with convergence orders O(τmin⁡{3−γ,2−β,1+γ−2β}+hx4+hy4) and O(τmin⁡{3−γ,2−β}+hx4+hy4), where γ and β are orders of two Caputo fractional derivatives, τ, hx and hy are the time and space

This article introduces an efficient numerical method for solving the Black-Scholes partial differential equation (PDE) that governs European options. The methodology employs the backward Euler scheme to discretize the time derivative and incorporates the non-symmetric interior penalty Galerkin method for handling the spatial derivatives. The study aims to determine optimal order error estimates in

Advection-diffusion-reaction (ADR) models describe transport mechanisms in fluid or solid media. They are often formulated as partial differential equations that are spatially discretized into systems of ordinary differential equations (ODEs) in time for numerical resolution. This paper investigates the performance of variable stepsize, semi-implicit, backward differentiation formula (VSSBDF) methods

Anisotropic fluids, such as nematic liquid crystals, can form non-spherical equilibrium shapes known as tactoids. Predicting the shape of these structures as a function of material parameters is challenging and paradigmatic of a broader class of problems that combine shape and order. Here, we consider a discrete shape optimization approach with finite elements to find the configuration of two-dimensional

In this paper, we explore efficient methods for discretized linear systems that arise from eddy current optimal control problems utilizing an all-at-once approach. We propose a novel low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. First, we reformulate the resulting discretized linear system into a matrix equation

In this work, we apply the local Petrov-Galerkin method based on radial basis functions to solving the two dimensional linear hyperbolic equations with variable coefficients subject to given appropriate initial and boundary conditions. Due to the presence of variable coefficients of the differential operator, special treatment is carried out in order to apply Green's theorem and derive the variational

Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known

The purpose of this article is to explore the superconvergence behavior of the first-order backward-Euler (BE) implicit/explicit fully discrete schemes for the unsteady incompressible MHD equations with low-order mixed finite element method (MFEM) by utilizing the scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) methods. Through dealing with linear terms in implicit format and nonlinear

The Hamilton-Jacobi(HJ) equation represents a class of highly nonlinear partial differential equations. Classical numerical techniques, such as finite element methods, face significant challenges when addressing the numerical solutions of such nonlinear HJ equations. However, recent advances in neural network-based approaches, particularly Physics-Informed Neural Networks (PINNs) and neural operator

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively

Fully discrete virtual element methods with second-order accuracy in temporal direction require to choose smaller time steps in order to maintain the higher accuracy provided by the spatial direction. To overcome this restriction higher order time stepping methods are needed. In this work the general Newmark scheme for temporal discretization is considered along with the virtual element discretization

Using the T-coercivity theory as advocated in Chesnel and Ciarlet (2013) [25], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when

A novel two-grid symmetric mixed finite element analysis is considered for semi-linear second order hyperbolic problem. To overcome the saddle-point problem resulted by the traditional mixed element methods, a new symmetric and positive definite mixed procedure is first introduced to solve semi-linear hyperbolic problem. Then the a priori error estimates both in L2 and Lp-norm senses are derived. Meanwhile

The paper presents the average radial particle method (ARPM), a new mesh-free technique for solving partial differential equations (PDEs). Here, we use the ARPM to solve 3D transient heat transfer problems. ARPM numerically approximates spatial derivatives by discretizing the domain by particles such that each particle is only affected by its direct neighbors. One feature that makes ARPM different

The paradigm of physics-driven forward electromagnetic computation holds significance for enhancing the accuracy of network approximations while reducing the dependence on large-scale datasets. However, challenges arise during the training process when dealing with objective functions characterized by high-frequency and multi-scale features. These challenges frequently occur as difficulties in effectively

This paper addresses the challenge of pricing options under stochastic volatility (SV) models, where explicit formulae are often unavailable and parameter estimation requires extensive numerical simulations. Traditional approaches typically either rely on large volumes of historical (option) data (data-driven methods) or generate synthetic prices across wide parameter grids (model-driven methods).

In this paper, we focus on the optimal control problem governed by a linear reaction-diffusion equation with constraints on the control variable. We construct an effective fully-discrete scheme to solve this problem by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the Galerkin spectral methods in space. By using the recently developed techniques

An accurate calculation of the traffic density is a key factor in understanding the formation and evolution of the traffic-related emission concentration in urban areas. We have developed a two-dimensional numerical model to solve traffic flow/pollution coupled problem whose pollution source is generated by the density of vehicles. The numerical solution of this problem is calculated via an algorithm

We develop Aubin–Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the heat source and initial datum are sufficiently smooth, we prove that the L2 error of approximations of the scalar field variable converges at a higher rate than the

We propose a time-dependent Advection Reaction Diffusion (ARD) N-species competition model to investigate the Stocking and Harvesting (SH) effect on population dynamics. For ongoing analysis, we explore the outcomes of a single species and competition between two competing species in a heterogeneous environment under no-flux boundary conditions, meaning no individual can cross the boundaries. We establish

Three-dimensional (3D) reconstruction from points cloud is an important technique in computer vision and manufacturing industry. The 3D volume consists of a set of voxels which preserves the characteristics of scattered points. In this paper, a 3D shell (narrow volume) reconstruction algorithm based on the Allen–Cahn (AC) phase field model is proposed, aiming to efficiently and accurately generate

In this paper, a linearized 2-step backward differentiation formula (BDF2) Galerkin method is proposed and investigated for Schrödinger equation with cubic nonlinearity and unconditionally optimal error estimate in L2-norm is obtained without any time-step restriction. The key to the analysis is to bound the H1-norm between the numerical solution and the Ritz projection of the exact solution by mathematical

When the radiation is in equilibrium with matter, a nonlinear parabolic equation is formed by the approximation of single temperature diffusion equation. In the actual numerical simulation, most of the time is used to solve the linear equations by the implicit discretization so as to retain the stability. In this paper, the discretization of the nonlinear diffusion equation on time is still full-implicit

In this study, a semi-implicit finite element method is proposed to solve the coupled system of infiltration and solute transport in porous media. The Richards equation is used to describe unsaturated flow, while the advection-dispersion equation (ADE) is used for modeling solute transport. The proposed approach is applied to linearize the system in time, avoiding iterative processes. A free parameter

This paper proposes a shape transformation model based on the Allen-Cahn equation, and its numerical scheme. The model overcomes the limitations of previous shape transformation models by introducing a convective term, realizing a smooth and stable shape transformation when the initial shape is not in contact with the target shape. To solve the problem of high-dimensions and the complexity of nonlinear

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