Compact Finite Difference Method Research Articles

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative methods that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity

Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.

A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method

<abstract><p>The 1D and 2D spatial compact finite difference schemes (CFDSs) for time-fractional diffusion equations (TFDEs) were presented in this article with uniform temporal convergence order. Based on the idea of the modified block-by-block method, the CFDSs with uniform temporal convergence order for TFDEs were given by combining the fourth-order CFDSs in space and the high order scheme in time. The stability analysis and convergence order of CFDSs with uniform convergence order in time for TFDEs strictly proved that the provided uniform accuracy time scheme is $ (3-\alpha) $ temporal order and spatial fourth-order, respectively. Ultimately, the astringency of 1D and 2D spatial CFDSs was verified by some numerical examples.</p></abstract>

Hydrothermal characteristics of ferrofluid in a wavy chamber with magnetic field-dependent viscosity: Effects of moving walls

The present investigation aims to scrutinize the role of different moving walls on the hydrothermal characteristics and thermal performance of a Fe3O4-H2O ferrofluid within a wavy chamber under mixed convection. Notably, the study also takes into consideration the effects of a magnetic field and internal heat generation or absorption phenomena. To achieve this, a numerical simulation employing a compact finite difference method is conducted, focusing on key factors such as the Richardson number (0.1≤Ri≤100), Hartmann number (0≤Ha≤30), magnetic orientation (00≤γ≤900), Grashof number (Gr=104), solid concentrations (0≤ϕnp≤0.04), magnetic number (0≤δ0≤1), Prandtl number (Pr=6.8377), and internal heat generation parameter (−2≤Q≤2). All of this is undertaken while carefully managing the ferromagnetic particle volume fraction. The primary aim is to assess how different types of moving walls affect the transport behavior of the ferrofluid and to determine the impacts of the magnetic field on the hydrothermal characteristics of the ferrofluid. The outcomes of this investigation reveal that the presence of moving walls leads to a non-uniform temperature distribution within the ferrofluid, significantly affecting the overall transport behavior. The addition of ferrofluid particles to the base fluid results in noteworthy enhancements in the average Nusselt numbers, signifying improved convective heat transfer efficiency. Specifically, Case-I experienced an increase of up to 3.21%, Case-II showed an improvement of 0.93%, Case-III exhibited a rise of 1.70%, and Case-IV demonstrated the highest enhancement of 20.71%. Furthermore, the inclusion of magnetic field-dependent viscosity and the internal heat generation coefficient significantly influences the behavior of the ferrofluid. The findings from this study hold paramount importance for the design and optimization of heat transfer systems that employ ferrofluids and have the potential to contribute to the development of novel technologies with a wide range of industrial applications.

A Compact Split-step Finite Difference Method for Solving the Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation arises from quantum mechanics and is extensively used in many fields of science and engineering. Thus, it is important to construct the high-order and stable numerical scheme of the Schrödinger equation. To solve the high-order and stable numerical solution of the nonlinear Schrödinger equation, the compact split-step finite difference method and the local one-dimensional method are combined in this paper. To attain high-order accuracy in time and space, the 4-order compact finite difference in space discretization is combined with the L-stable Simpson method in time discretization. Therefore, a scheme with 4-order accuracy in space and 3-order accuracy in time is obtained, and the stability of the scheme is analyzed. Finally, numerical results manifest that the devised scheme can supply accurate and stable results to the nonlinear Schrödinger equation.

Vortex dynamics of accelerated flow past a mounted wedge

This study is concerned with the simulation of a complex fluid flow problem involving flow past a wedge mounted on a wall for channel Reynolds numbers Rec = 1560, 6621, and 6873 in uniform and accelerated flow medium. The transient Navier–Stokes (N–S) equations governing the flow have been discretized using a recently developed second order spatially and temporally accurate compact finite difference method on a nonuniform Cartesian grid by the authors. Almost all the flow characteristics of a well-known laboratory experiment of Pullin and Perry [“Some flow visualization experiments on the starting vortex,” J. Fluid Mech. 97(2), 239–255 (1980)] have been captured by our numerical simulation, and we provide a qualitative and quantitative assessment of the same. Furthermore, the influence of the parameter m, controlling the intensity of acceleration, has been discussed in detail along with the intriguing consequence of non-dimensionalization of the N–S equations pertaining to such flows. The simulation of the flow across a time span significantly greater than the aforesaid lab experiment is the current study's major achievement. Meanwhile, a grid independence study performed in the process confirmed that our simulation is devoid of any under-resolution or numerical artifact. For the accelerated flow, the onset of shear layer instability leading to a more complicated flow toward transition to turbulence has also been aptly resolved. The quality of our simulation is validated by the close similarity of our simulation to the high Reynolds number experimental results of Lian and Huang [“Starting flow and structures of the starting vortex behind bluff bodies with sharp edges,” Exp. Fluids 8(1–2), 95–103 (1989)] for the accelerated flow across a typical flat plate. All three steps of vortex shedding, including the exceedingly intricate threefold structure, have been captured quite efficiently.

A compact high resolution semi-variable mesh exponential finite difference method for non-linear boundary value problems of elliptic nature

In this research an original exponential approximation of second accuracy in y- and third accuracy in x-axis employing full step discretization has been designed for solving 2D non-linear partial differential equation of elliptic nature in a rectangular domain. We adopted non-constant grid spacing in x -axis and constant grid spacing in y -axis in numerical computation of convection-diffusion equation where convection term dominates. An exhaustive error behaviour of the technique has been analysed. Non-linear elliptic equations are computed using this method. Lastly, proposed idea is scrutinized on simulations of physical repute with emphasis on convection-diffusion equation articulating the efficacy of the technique.

Thermogravitational Convection in a Multiple Baffled Enclosure Filled with Magneto-Hybrid Nanofluid Subjected to Magnetic Field Dependent Viscosity

This study explores the significant impacts of thin baffles and magnetic field dependent viscosity on magnetohydrodynamic (MHD) thermogravitational convection of Cu-Al2O3 (50%–50%) water hybrid nanoliquid in a cavity. Considering different arrangements of baffle sticks on both the vertical walls, four geometrical configurations (Case-I, Case-II, Case-III and Case-IV) have been analyzed. Numerical simulation has been performed for the governing Navier-Stokes (N-S) equations in streamfunction - vorticity form having energy equation. These coupled equations are solved by proposing a higher-order compact finite difference method. The combination of five important aspects (hybrid nanofluid, multiple baffles, magnetic field dependent viscosity (MFDV), magnetic field and compact computation) signifies the novelty of this work. Fluid flow and transportation of thermal energy within the stipulated domain are presented for various flow pertinent parameters. The outcomes show that the increase in number of baffles diminishes the average Nusselt number values. It is concluded here that an increase in Hartmann number from 0 to 90 leads to a decrease in average Nusselt number up to 23.7% for Case-I, 23.8% for Case-II, 21.2% for Case-III and 28% for Case-IV in presence of MFDV effects.

Deep FDM: Enhanced finite difference methods by deep learning

In this work, we propose a new idea to improve numerical methods for solving partial differential equations (PDEs) through a deep learning approach. The idea is based on an approximation of the local truncation error of the numerical method used to approximate the spatial derivatives of a given PDE. We present our idea as a proof of concept to improve the standard and compact finite difference methods (FDMs), but it can be easily generalized to other numerical methods.Without losing the consistency and convergence of the FDM numerical scheme, we achieve a higher numerical accuracy in the presented one- and two-dimensional examples, even for parameter ranges outside the trained region. We also perform a time complexity analysis and show the efficiency of our method.

Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

An efficient numerical scheme and its analysis for the multiterm time‐fractional convection‐diffusion‐reaction equation

This article aims at developing a robust numerical method based on graded mesh for solving the multiterm time‐fractional convection‐diffusion‐reaction (TFCDR) equation whose solution very likely exhibits a weak singularity at the initial time. The time‐fractional derivative in the model problem is described in terms of Liouville–Caputo. In order to handle the weak singularity at the initial time, we use a graded mesh technique for discretization of multiterm temporal fractional derivatives. The space derivatives are approximated by means of a compact finite difference (CFD) method. The proposed method is analyzed for its stability and convergence. Two numerical examples are considered to demonstrate the applicability and accuracy of the method. It is shown that the proposed graded mesh technique provides an optimal rate of convergence in time direction for the problem with nonsmooth exact solution, while the method on the uniform mesh yields a nonoptimal rate of convergence.

A compact ADI finite difference method for 2D reaction–diffusion equations with variable diffusion coefficients

Reaction–diffusion systems on a spatially heterogeneous domain have been widely used to model various biological applications. However, solving such partial differential equations (PDEs) analytically is rarely possible. Therefore, efficient and accurate numerical methods for solving such PDEs are desired. In this paper, we apply the well-known Padé approximation-based operator splitting techniques and develop a fourth-order compact alternative directional implicit (ADI) scheme. The new scheme is compact and fourth-order accurate in space. Combined with the Richardson extrapolation, the method can be improved to fourth-order accuracy in time. Stability analysis shows that the method is unconditionally stable; thus, a large time step can be used to improve the overall computational efficiency. Numerical examples have also demonstrated the new scheme’s high efficiency and high order accuracy.