A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming EQ1rot element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken H1-norm is achieved directly by the energy dissipation property without using the known time-space splitting technique in the existing

A high-order implicit–explicit (IMEX) finite element method with energy conservation is constructed to solve the regularized logarithmic Schrödinger equation (RLogSE) with a periodic boundary condition. The discrete scheme consists of the relaxation-extrapolated Runge–Kutta (RERK) method in the temporal direction and the finite element method in the spatial direction. Choosing a proper relaxation parameter

In this manuscript, we study a generalized fractional reaction-diffusion model involving a distributed-order operator. An efficient hybrid approach is proposed to solve the presented model. The L1 approximation is utilized to discretize the time variable, while the mixed finite element method is employed for spatial discretization. A detailed error and stability analysis of the proposed method is provided

Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on

In this article, we present a numerical scheme for solving a time-fractional stochastic Cahn–Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order α∈(0,1) and a fractional time-integral noise of order γ∈[0,1]. Our numerical approach combines a piecewise linear finite element method in space with a convolution

The main purpose of this study is to develop a numerical model of unsaturated flow in soils with plant root water uptake. The Richards equation and different sink term formulations are used in the numerical model to describe the distribution of soil moisture in the root zone. The Kirchhoff transformed Richards equation is used and the Gardner model is considered for capillary pressure. In the proposed

This paper presents a high-precision numerical method based on spectral deferred correction (SDC) for solving the time-fractional Allen-Cahn equation. In the temporal direction, we establish a stabilized variable-step L1 semi-implicit scheme which satisfies the discrete variational energy dissipation law and the maximum principle. Through theoretical analysis, we prove that this numerical scheme is

The high-order gas-kinetic scheme (HGKS) with WENO spatial reconstruction method has been extensively validated through numerous numerical experiments, demonstrating its superior accuracy, efficiency and robustness. In comparison to WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. This paper introduces two

Isogeometric Analysis (IgA) is a spline-based approach to the numerical solution of partial differential equations. The concept of IgA was designed to address two major issues. The first issue is the exact representation of domains generated from Computer-Aided Design (CAD) software. In practice, this can be realized only with multi-patch IgA, often in combination with trimming or similar techniques

The conventional finite element method (FEM) usually fails to generate sufficiently fine numerical solutions in the analyses of Mageto-electro-elastic (MEE) structures in which three different types of physical fields are coupled together. To enhance the performance of the FEM in analyzing MEE structures, in this work a novel overlapping triangular finite element is introduced for dynamic analysis

This paper presents an effective nonlocal finite element method (FEM) for investigating the free vibration behavior of sigmoid functionally graded material (S-FGM) nanoshells using nonlocal elasticity theory. The effective nonlocal parameters via third-order shear deformation theory (TSDT) are varied along the thickness of the nanoshells following the sigmoid function. In this study, two different

In this paper, we study a posteriori error estimates of the fast L1−2 scheme for time discretization of time fractional parabolic differential equations. To overcome the huge workload caused by the nonlocality of fractional derivative, a fast algorithm is applied to the construction of the L1−2 scheme. Employing the numerical solution obtained by the fast L1−2 scheme, a piecewise continuous function

We study the numerical evaluation of the integral fractional Laplacian and its application in solving fractional diffusion equations. We derive a pseudo-spectral formula for the integral fractional Laplacian operator based on fractional order-dependent, generalized multi-quadratic radial basis functions (RBFs) to address efficient computation of the hyper-singular integral. We apply the proposed formula

A system of two coupled nonlinear parabolic partial differential equations with two opposite directions of time is considered. In fact, this is the so-called “Mean Field Games System” (MFGS), which is derived in the mean field games (MFG) theory. This theory has numerous applications in social sciences. The topic of Coefficient Inverse Problems (CIPs) in the MFG theory is in its infant age, both in

The paper develops the essentially optimal sparse tensor product finite element method for solving two scale elliptic and parabolic problems in a domain D⊂Rd, d=2,3, which is embedded with a periodic array of inclusions of microscopic sizes and spacing. The two scale coefficient is thus discontinuous in the fast variable. We obtain approximations for the solution of the homogenized equation and the

This paper considers the oscillatory flow of the Oldroyd-B fluid in a tube with a right triangular cross-section. The partial differential equation for describing the unidirectional flow of Oldroyd-B fluid is derived. For treating the triangular region, the unstructured mesh finite element method is applied. For verifying the accuracy of the finite element method, the source term is introduced and

Cubature, i.e. multivariate numerical integration, plays a core part in the finite-element method. For a given element geometry and interpolation, it is possible to choose different cubature rules, leading to concepts like full and reduced integration. These cubature rules are usually chosen from a rather small set of existing rules, which were not specifically derived for finite-element applications

This paper proposes a numerical method for solving the time-fractional Stokes-Darcy problem using a partitioned time stepping algorithm with the Beavers-Joseph-Saffman condition. The stability of the method is established under a moderate time step restriction, τ≤C where C represents physical parameters, by utilizing a discrete fractional Gronwall type inequality. Additionally, error estimates are

The partial differential equation describing equilibrium radiation diffusion is strongly nonlinear, which has been widely utilized in various fields such as astrophysics and others. The equilibrium radiation diffusion equation usually appears over multiple complicated domains, and the material characteristics vary between each domain. The diffusion coefficient near the interface is discontinuous. In

This work presents an accurate in-plane vibration analysis of functionally graded material (FGM) skew plates with elastically restrained boundaries using the variational differential quadrature method (VDQM). The weak form of the governing equations is derived by integrating two-dimensional elasticity theory with Hamilton's principle. The differential and integral operators are directly converted into

This paper focuses on the design of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behavior of the solution. The multi-term Caputo time fractional derivatives

A semi-analytic numerical method is described as an efficient meshless approach for the solution of anomalous non-linear thermal conduction problems in functionally graded materials in which the model results in fractional boundary value problems. The first key feature in this scheme is the derivation and discretization of the fractional derivative at every time step. The second key feature is the

This paper is devoted to the extension of a quasi-Newton/variable preconditioning (QNVP) method to non-smooth problems, motivated by elasto-plastic models. Two approaches are discussed: the first one is carried out via regularized approximations of the nonsmooth problem, and the second one gives an extension to nonsmooth operators in order to be applied directly. Convergence analysis is presented for

In this paper, an energy stable bound-preserving finite volume scheme is constructed for the Allen-Cahn equation. The first-order operator splitting method is used to split the original equation into a nonlinear equation and a heat equation in each time interval. The nonlinear equation is solved by the explicit scheme, and the heat equation is discretized by the extremum-preserving scheme. The harmonic

In this paper, a modified scaled boundary finite element method (SBFEM) is developed to study static and vibration behaviors of laminated conical shells under the conical coordinate system. In the modified SBFEM, the geometry of the conical shell is defined entirely by scaling the internal surface of the structure. This approach eliminates geometric errors caused by discretization, thereby enhancing

This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover

We present a mathematical model describing the equilibrium of a viscoelastic body with long-term memory in frictional contact with a sliding foundation. The process is quasistatic, and material damage resulting from excessive stress or strain is captured by a damage function. We assume the material is inhomogeneous, leading to multiple contact boundary conditions. The contact interface is partitioned

In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form I−A(σ). The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and

In this study, we propose two different approaches of a two-step method for solving a system combining the heat and the wave equations. Our focus centers on the transmission problem in two dimensions, with a primary objective of numerically characterizing the distribution of temperature and pressure. First we apply a semi-discretization with respect to time by using the Laguerre transformation. This

We analyze the first order Enlarged Enhancement Virtual Element Method (E2VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement property (from which comes the prefix E2) of local virtual spaces. We provide

As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application,

A semi-Lagrangian radial basis function partition of unity (RBF-PU) closest point method is designed for solving advection-diffusion equations on surfaces. This new meshfree method combines the semi-Lagrangian method with the RBF-PU closest point method. The semi-Lagrangian RBF-PU closest point method traces the departure point backwards along the velocity field based on patches at each time step.

The article is concerned with and analyzes the α-robust error bound for time-fractional diffusion wave equations with weakly singular solutions. Nonuniform L1-type time meshes are used to handle non-smooth systems, and the sum-of-exponentials (SOEs) approximation for the kernels function is adopted to reduce the memory storage and computational cost. Meanwhile, the virtual element method (VEM), which

In this paper, the nonlocal electron heat transport model in one and two dimensions is considered and studied. An energy stability finite element method is designed to discretize the nonlocal electron heat transport model. For the nonlinear discrete system, both Newton iteration and implicit-explicit (IMEX) schemes are employed to solve it. Then the energy stability is proved in semi-discrete and fully-discrete

In this paper, one-dimensional and two-dimensional pure-positivity-preserving (PPP) methods are proposed for fifth-order finite volume multi-resolution WENO (MR-WENO) schemes to solve some extreme problems on structured meshes. The MR-WENO spatial reconstruction procedures only require one five-cell, one three-cell, and one one-cell stencils for achieving uniform fifth-order accuracy in smooth regions

We construct a fourth-order accurate compact finite difference scheme that is well-balanced, positivity-preserving of water height, and bound-preserving of temperature for Ripa and concentration for pollutant transport systems. The proposed scheme preserves the still-water steady state and the positivity of water height. It also maintains concentration bounds for pollutants across nonflat bottom topographies

The deep mixed residual method (DeepMRM) is a technique to solve partial differential equation. In this paper, it is applied to tackle PDE-constrained optimization problems (PDE-COPs). For a PDE-COP, we transform it into an optimality system, and then employ mixed residual method (MRM) on this system. By implementing the DeepMRM with three different network structures (fully connected neural network

Fluid-structure interaction (FSI) problems with strong nonlinearity and multidisciplinarity pose challenges for current numerical FSI algorithms. This work proposes a monolithic strategy for solving the equations of motion for both the fluid and structural domains under the unique Lagrangian framework of the B-spline material point method (BSMPM). A node-based density smoothing BSMPM (referred to as

Compared to the classical wave equation, the nonlocal wave equation incorporates a nonlocal operator and can capture a broader range of practical phenomena. However, this nonlocal formulation significantly increases the computational cost in numerical simulations, necessitating the development of efficient and accurate time integration schemes. Inspired by the newly developed generalized scalar auxiliary

In this paper, a particle method is used to approximate the solutions of a “fluid-like” macroscopic traffic flow model for automated vehicles. It is shown that this method preserves certain differential inequalities that hold for the macroscopic traffic model: mass is preserved, the mechanical energy is decaying and an energy functional is also decaying. To demonstrate the advantages of the particle

In this paper, we mainly focus on the numerical approximations of the electro-hydrodynamic system, which couples the Poisson-Nernst-Planck equations and the Navier-Stokes equations. A novel linear, fully-decoupled and energy-stable finite element scheme for solving this system is proposed and analyzed. The fully discrete scheme developed here is employed by the stabilizing strategy, implicit-explicit

A class of mathematical models are proposed for modelling and numerical simulation of coupled radiative and conductive heat transfer in non-grey absorbing and emitting media under phase change. Progress in this area of mathematical modelling would contribute to a sustainable future manufacturing involving high temperature and phase change. Accurately predicting phase-change interface is the crucial

In order to solve the vibration and acoustic characteristics of spherical shell in light fluid environment, based on Jacobi-Ritz-spectral BEM, a unified analysis formula for acoustic vibration of spherical shell under arbitrary boundary conditions is established. Based on the First-order shear deformation theory (FSDT) and domain decomposition method (DDM), the theoretical model of spherical shell

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations

The behaviour of non-Newtonian fluids, and their interaction with other fluid phases and components, is of interest in a diverse range of scientific and engineering problems. In the context of the lattice Boltzmann method (LBM), both non-Newtonian rheology and multiphase flows have received significant attention in the literature. This study builds on that work by presenting the development and validation

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on

We consider the discretization of the 1d-integral Dirichlet fractional Laplacian by hp-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of hp-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial

Besides diffusion and capillary, convection which is unavoidable under terrestrial condition has remarkable effects on the microstructure evolution during solidification. In this study, a phase-field lattice-Boltzmann model, accelerated by state-of-the-art parallel-adaptive mesh refinement algorithm, is solved to investigate the morphological evolution of the Mg-Gd dendrite under convection. The lengths

In this work, a topology optimization model for designing devices that operate with multiple relative velocities considering turbulent compressible flows is proposed. The model consists of the Favre-averaged Navier-stokes equations in an axisymmetric domain coupled with a continuous boundary propagation model. The propagation is used to impose different solid behaviors based on which wall it is connected

Cerebral aneurysms represent a life-threatening condition associated with considerable morbidity and mortality rates. The formation of cerebral aneurysms is influenced by various factors, including vessel geometry, blood flow characteristics, and hemodynamic forces. In this study, we investigate the impact of vessel geometry on the formation of cerebral aneurysms utilizing computational fluid dynamics

In the present work, a new methodology is proposed for building surrogate parametric models of engineering systems based on modular assembly of pre-solved modules. Each module is a generic parametric solution considering parametric geometry, material and boundary conditions. By assembling these modules and satisfying continuity constraints at the interfaces, a parametric surrogate model of the full

This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems on general unstructured meshes with mixed boundary conditions. By leveraging the Implicit Layer Theory, Ψ-GNN models an “infinitely” deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly

This paper develops a robust and efficient PDE-constrained optimal control model for multicomponent pollutions in porous media, which takes into account nonlinear multi-component contamination flows of groundwater. The objective of the pollution optimal control is to identify the optimal injection rates from the top part boundary of domain, which can minimize the least squares error between the concentrations

The discrete velocity method (DVM) for rarefied flows and the unified methods (based on the DVM framework) for flows in all regimes, from continuum one to free molecular one, have worked well as precise flow solvers over the past decades and have been successfully extended to other important physical fields. Both DVM and unified methods endeavor to model the gas-gas interaction physically. However

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering

In this paper, we propose a nonconforming extended virtual element method, which combines the extended finite element method with the nonconforming virtual element method, for solving Stokes interface problems with the unfitted-interface mesh. By introducing some stabilization terms and penalty terms, as well as some special terms defined on non-cut edges of interface elements in the discrete bilinear

Nonlinear time-fractional diffusion problems, a significant class of parabolic-type problems, appear in various diffusion phenomena that seem extensively in nature. Such physical problems arise in numerous fields, such as phase transition, filtration, biochemistry, and dynamics of biological groups. Because of its massive involvement, its accurate solutions have become a challenging task among researchers