Uniform Mesh Research Articles

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

ABSTRACTUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method’s applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.

Extended B-Spline Collocation Method for Singularly Perturbed Time Delay Parabolic Differential-Difference Problems

The paper presents a robust collocation B-spline numerical scheme for solving singularly perturbed time lag parabolic differential–difference problems. The method uses Taylor’s series expansion for the approximation of retarded terms, the [Formula: see text]-method (when [Formula: see text]) for time discretization, and the extended third-degree B-spline collocation method for spatial discretization in a piecewise uniform Shishkin mesh. The advantage of using an extended cubic B-spline rather than the classical third-degree B-spline method is that it introduces one additional free parameter to control the global shape parameter. The stability and uniform convergence of the proposed scheme are investigated. The method is first-order accurate in [Formula: see text] and almost second-order accurate in [Formula: see text], surpassing existing literature methods.

A new high-order RKDG method based on the TENO-THINC scheme for shock-capturing

In recent years, Runge-Kutta Discontinuous Galerkin (RKDG) methods have gained substantial attention in solving hyperbolic conservation laws, attributed to their high-order accuracy and adaptability to unstructured meshes. However, standard RKDG methods cannot capture discontinuities without oscillation unless they are supplemented with troubled cell indicators and limiters. Existing indicators, such as the total variation bounded (TVB) minmod indicator and the KXRCF indicator, typically depend on a critical parameter tied to the equation's solution, necessitating tuning for different cases. In terms of limiters, the popular limiters even fail to guarantee the high-order property in the smooth regions. The advent of Weighted Essentially Non-Oscillatory (WENO) schemes prompts the implementation of WENO limiters for RKDG methods, preserving high-order properties. However, WENO-family schemes exhibit significant numerical dissipation, potentially smearing small-scale flow structures. In this work, the Targeted Essentially Non-Oscillatory (TENO) indicator [1] is utilized, which leverages the nonlinear weighting strategy of the TENO scheme to separate high-wavenumber physical fluctuations and genuine discontinuities from smooth regions with a unified set of parameters. For troubled cells, a novel limiter is proposed for structured meshes, which combines a TENO scheme for resolving high-wavenumber physical fluctuations and a novel non-polynomial Tangent of Hyperbola for the INterface Capturing (THINC) scheme for resolving genuine discontinuities with extremely low numerical dissipation. Furthermore, the shifting between the TENO and THINC schemes is based on a new boundary variation diminishing (BVD) strategy, which only relies on compact neighborhoods and is significantly simpler than its predecessors. Meanwhile, a new strategy is proposed to ensure the consistency of the new limiter applied for 1D and 2D cases. A set of 1D and 2D benchmark cases including strong shockwaves and a broad range of flow length scales is simulated with uniform meshes for 1D cases and structured quadrilateral meshes for 2D cases to demonstrate the performance of the new numerical scheme. The indicator does not activate any limiters in the accuracy test cases to ensure the high-order property of the whole numerical scheme. Other cases with discontinuities demonstrate the low-dissipation properties of the new limiter in comparison to the simple WENO limiter.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

The Influence of Local Constraints on Solvent Motion in Polymer Materials.

The influence of obstacles in the form of polymer chains on the diffusion of a low-molecular-weight solvent was the subject of this research. Studies were performed by computer simulations. A Monte Carlo model-the Dynamic Lattice Liquid algorithm-based on the idea of cooperative movements was used. The tested materials were polymer networks with an ideal structure (with a uniform mesh size) and real, irregular networks (with a non-uniform mesh size) obtained numerically by copolymerization. The diffusion of the solvent was analyzed in systems with a polymer concentration that did not exceed 16%. The influence of the polymer concentration and macromolecular architecture structure on the mobility and character of the motion of the solvent was discussed. The influence of irregular network morphology on solvent dynamics appeared to be significantly stronger than that of regular networks and star-like polymers.

A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh

In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.

A parameter-uniform hybrid method for singularly perturbed parabolic 2D convection-diffusion-reaction problems

The solution of the singular perturbation problems (SPP) of convection-diffusion-reaction type may exhibit regular and corner layers in a rectangular domain. In this work, we construct and analyze a parameter-uniform operator-splitting alternating direction implicit (ADI) scheme to efficiently solve a two-dimensional parabolic singularly perturbed problem with two positive parameters. The proposed model is a combination of the backward-Euler method defined on a uniform mesh in time and a hybrid method in space defined on a special Shishkin mesh. The analysis is presented on a layer adapted piecewise-uniform Shishkin mesh. The developed numerical method is proved to be first-order convergent in time and almost second-order convergent in space. The numerical experiments are performed to validate the theoretical convergence results and illustrate the efficiency of the current strategy.

Finite Element Method on locally refined composite meshes for Dirichlet fractional Laplacian

It is known that the solution of the Dirichlet fractional Laplacian in a bounded domain exhibits singular behavior near the boundary. Consequently, numerical discretizations on quasi-uniform meshes lead to low accuracy and nonphysical solutions. We adopt a finite element discretization on locally refined composite meshes, which consist in a combination of graded meshes near the singularity and uniform meshes where the solution is smooth. We also provide a reference strategy on parameter selection of locally refined composite meshes. Numerical tests confirm that finite element method on locally refined composite meshes has higher accuracy than uniform meshes, but the computational cost is less than that of graded meshes. Our method is applied to discrete the fractional-in-space Allen–Cahn equation and the fractional Burgers equation with Dirichlet fractional Laplacian, some new observations are discovered from our numerical results.

Routing impact of architecture and damage in programmable photonic meshes

Programmable photonic integrated circuits (PPICs) emerge as a novel technology with an enormous potential for ground-breaking innovation. Different architectures are currently being considered that dictate how waveguides should be connected to realize a broadly usable circuit. We focus on the effect of varying connectivity architectures on the routing of light. Three types of uniform meshes are studied, and we introduce a newly developed mesh that is called ring-connected straight lines. We provide an analytical formula to calculate exact distances in these meshes and introduce several metrics relating to routing to compare these meshes. We show that hexagonal tiles are the most promising, but the ring-connected straight lines architecture has a use case as well. Besides this, the effect of defect couplers is also studied. We find that the effects of these failures vary greatly by type and severity on the routability of the mesh.

Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η∈(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2−η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 &lt; ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

A SUPS formulation for simulating unsteady natural/mixed heat convection phenomena in square cavities under intense magnetic forces

This paper presents stabilized finite element simulations of unsteady natural and mixed heat convection phenomena occurring in square enclosures. It is assumed that the enclosures are subject to intense uniform magnetic fields applied perpendicular to the vertical walls. It is well known that the classical Galerkin finite element method (GFEM) is prone to yielding nonphysical oscillations when simulating convection-dominated flows. Therefore, in order to handle such flows occurring at high Hartmann (0≤Ha≤100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le {\\rm Ha} \\le 100$$\\end{document}) and Rayleigh (103≤Ra≤108\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{3} \\le {\\rm Ra} \\le 10^{8}$$\\end{document}) numbers, the stabilization of the GFEM is accomplished with the streamline-upwind/Petrov–Galerkin and pressure-stabilizing/Petrov–Galerkin formulations. Temporal discretizations are performed with the backward Euler formula. At each time step, the nonlinear systems are linearized with the Newton–Raphson (N–R) method, and the linearized systems are solved directly using the sparse lower–upper decomposition. The incompressible flow solvers are developed in-house, run in parallel within the FEniCS ecosystem, and tested on a comprehensive set of numerical experiments with various thermal boundary conditions. Numerical simulations and comparisons reveal that the proposed formulation performs quite well at high Hartmann and Rayleigh numbers without exhibiting any significant localized or globally spread numerical instabilities. Moreover, it achieves this by using uniform meshes and only linear elements, making the formulation much more computationally efficient. Mesh convergence analyses are also performed to demonstrate the reliability of the numerical results obtained.

Two Schemes Based on the Collocation Method Using Müntz–Legendre Wavelets for Solving the Fractional Bratu Equation

Our goal in this work is to solve the fractional Bratu equation, where the fractional derivative is of the Caputo type. As we know, the nonlinearity and derivative of the fractional type are two challenging subjects in solving various equations. In this paper, two approaches based on the collocation method using Müntz–Legendre wavelets are introduced and implemented to solve the desired equation. Three different types of collocation points are utilized, including Legendre and Chebyshev nodes, as well as uniform meshes. According to the experimental observations, we can confirm that the presented schemes efficiently solve the equation and yield superior results compared to other existing methods. Also, the schemes are convergent.

Second-order error analysis of a corrected average finite difference scheme for time-fractional Cable equations with nonsmooth solutions

In this paper, we first formulate an average finite difference scheme with appropriate correction terms for the time-fractional initial-value problem of y′(t)+0RLDt1−αy(t)=g(t), where 0RLDt1−α denotes the Riemann–Liouville fractional derivative of order 1−α(α∈(0,1)). It is shown that, under some suitable conditions on the data, the numerical solution converges to the exact solution with order O(τ2), where τ is the time step size. The method is then extended to solve the time-fractional Cable equation, combined with a standard discretisation of the spatial derivatives on a uniform mesh. The second-order time convergence rate is proved. Several numerical examples are given to illustrate the good agreement with the theoretical analysis of the presented methods.

Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order α with α ∈ (0, 1) is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time t = 0. Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the temporal fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate (2 − α) for suitable choice of grading parameter (γ ≥ (2 − α)/α). Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for γ ≥ 2/α. In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

Second-order numerical method for a neutral Volterra integro-differential equation

This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.