Standard Finite Difference Method Research Articles

A new numerical approach to Gardner Kawahara equation in magneto‐acoustic waves in plasma physics

AbstractThe basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg‐de Vries equation, by means of finite element method. For this purpose, a collocation finite element method based on trigonometric quintic B‐spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank–Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B‐spline basis functions. More specifically, the error norms and are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.

Iterative compact finite difference method for the numerical study of fully wet porous fins with different profile shapes

This paper investigates the temperature distribution of porous fins with different profiles (rectangular, triangular, convex parabolic and concave parabolic profiles) under fully wet conditions. The porous fin material is Al and the tips are insulated. An iterative discrete method is employed to solve the nonlinear governing equations. For this purpose, the governing equations are converted into a sequence of linear differential equations by the quasi-linearization method. Then, the numerical solutions of these linear equations are estimated using the compact finite difference scheme. The main advantage of the method is that it does not require to solve the system of non-linear algebraic equations. Moreover, unlike the Padé standard finite difference method, there is no need to calculate the derivatives of the solution from the governing differential equation in order to obtain the finite difference formulas with higher accuracy. The convergence analysis of the method is studied in detail. The effect of geometric and thermophysical parameters on temperature distribution and fins' efficiency is discussed using the plotted graphs. Comparing our numerical solutions with those obtained by the least squares method and the Chebfun package in Matlab shows the accuracy and efficiency of the proposed method.

An [formula omitted]-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution

A fast algorithm is proposed for solving two-dimensional distributed-order time–space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss–Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be α-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz–Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.

Modeling One Dimensional Two-Cell Model with Tumor Interaction Using Krylov Subspace Methods

A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.

On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

<p style='text-indent:20px;'>We consider an initial and Dirichlet boundary value problem for a logarithmic Schrödinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete <inline-formula><tex-math id="M1">\begin{document}$ L_t^{\infty}(L_x^2) $\end{document}</tex-math></inline-formula> norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrödinger equation is provided, without regularizing its nonlinear term.</p>

Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation

<abstract><p>A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} u(x, y, t)- \Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.</p></abstract>

On the nonstandard finite difference method for reaction–diffusion models

The nonstandard finite difference (NSFD) method is an elegant approach in that it overcomes the numerical instability and bias exhibited by standard finite difference methods for numerically solving nonlinear differential equations. In addition, the NSFD method preserves some qualitative features of the continuous-time model such as boundedness and positivity. But for an important class of models which include diffusion and reaction–diffusion systems that appear in a number of application domains including epidemiology, ecology, and finance, recently introduced NSFD schemes do not guarantee first-order temporal accuracy or consistency. In this paper, we first show this for a reaction–diffusion epidemic model. We then propose an alternative NSFD scheme that guarantees first-order accuracy in time and second-order accuracy in space whilst preserving positivity of the solution. Stability and consistency analyses of the proposed scheme are then presented. To demonstrate the performance of the proposed scheme we show a comparison with the existing NSFD approach for three examples which confirm the superiority of our proposed approach.

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

AbstractIn this paper we introduce a new type of explicit numerical algorithm to solve the spatially discretized linear heat or diffusion equation. After discretizing the space variables as in standard finite difference methods, this novel method does not approximate the time derivatives by finite differences, but use three stage constant‐neighbor and linear neighbor approximations to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee unconditional stability. The scheme contains a free parameter p. We show that the convergence of the method is third‐order in the time step size regardless of the values of p, and, according to von Neumann stability analysis, the method is stable for a wide range of p. We validate the new method by testing the results in a case where the analytical solution exists, then we demonstrate the competitiveness by comparing its performance with several other numerical solvers.

Phase-field computations of anisotropic ice crystal growth on a spherical surface

In this paper, we present a numerical method for the phase-field model of anisotropic ice crystal growth on a spherical surface. The mathematical model includes terms related to the anisotropic interfacial energy, which is defined by the interface angle with respect to a reference angle. One of the natural numerical methods on curved surfaces is a computational technique based on a triangular mesh for the surface in a three-dimensional space. However, it is difficult to compute terms with the interface angle on a triangular mesh. To resolve this problem, we solve the governing equation in Cartesian coordinates after rotating each vertex and the 1-ring neighborhood of the vertex on the triangular mesh. After rotation and interpolation, we numerically solve the phase-field model using a standard finite difference method. We present the results of several tests to demonstrate that the proposed algorithm can recover anisotropic ice crystal growth on a spherical surface.

An efficient numerical scheme and its stability analysis for a time-fractional reaction diffusion model

An efficient numerical scheme is presented to solve a time-fractional reaction–diffusion (TFRD) model with Caputo temporal–fractional derivative. The L1 scheme is employed to approximate the Caputo temporal–fractional derivative and a collocation method based on sextic B-spline (SBS) basis functions is used for discretization of the spatial variable. The stability of the method is investigated. Three numerical examples are provided to illustrate the performance and accuracy of the method. The convergence order of proposed method is O(Δt2−α+Δx6), where α is the order of fractional derivative (FD) (0<α<1) and Δt and Δx represent, respectively, the temporal step size and the spatial mesh size. The influence of the order of fractional derivative α on the solution profile is examined. The computed results and CPU times are compared with those obtained by the standard second-order finite difference method.

Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

In this paper, different numerical schemes are developed and investigated for solving singularly perturbed delay differential equations (DDEs). The solution of the considered equations exhibits a strong boundary layer when the perturbation parameter approaches zero. Standard numerical methods developed for solving regular problems fail to treat accurately the considered equations. We developed three numerical schemes for treating the considered equations. The first scheme (Scheme I) uses a non-standard mid-point upwind finite difference method on uniform mesh; the second scheme (Scheme II) uses a standard mid-point upwind finite difference method on Shiskin mesh and the third scheme (Scheme III) uses a non-standard mid-point upwind finite difference method on Shiskin mesh. The existence of a unique discrete solution is investigated using the discrete maximum principle. The stability and uniform convergence of the schemes are investigated and proved. Scheme III gives better accuracy and order of convergence than Scheme I. It has a boundary layer resolving behaviour which is the main drawback of Scheme II. Numerical examples are considered and treated to validate the theoretical finding of the schemes for different values of the perturbation parameter and delay parameter.

Discretization Corrected Particle Strength Exchange for Steady State Linear Elasticity

Discretization corrected particle strength exchange (DC PSE) is a particle based spatial differential operator designed to solve meshless continuum mechanics problems. DC PSE is a spatial gradient operator that can discretize a computational domain with randomly distributed particles, provided that each particle has enough neighboring particles. In contrast, conventional methods such as the standard finite difference method require the computational domain to be discretized into a Cartesian grid. In linear elasticity simulations, especially steady state cases, this domain is mostly discretized using mesh-based methods such as finite element. However, while particle methods such as smoothed particle hydrodynamics (SPH) have been widely applied to solve dynamic elasticity problems, they have rarely been used in steady state simulations. In this study, a DC PSE operator was used to solve steady linear elasticity problems in a two-dimensional domain. The result of the DC PSE numerical simulation was compared to numerical results, empirical formula results, and results from conventional commercial finite element software, respectively.

Numerical prediction of vortex-induced vibrations of a long flexible riser with an axially varying tension based on a wake oscillator model

A numerical study based on the wake oscillator model has been conducted to determine the vortex-induced vibration (VIV) responses of a flexible riser with an axial time-varying tension. Three different types of flows, viz., linear shear, exponential shear, and real stepped flows, have been considered. The coupling equations of a structural oscillator and a wake oscillator have been solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, displacement envelope, and displacement evolution for three different flow profiles have been systematically compared. Subsequently, for each flow profile, the effect of the different tension models on the VIV characteristics has been investigated. The numerical results indicated that, both the VIV displacement and VIV frequency in real stepped flow had diverse characteristics from those in linear shear flow. Compared with the corresponding VIV characteristics in real stepped flow, the VIV displacement in exponential shear flow was similar, however, the VIV frequency in exponential shear flow was disparate. The VIV displacement with the constant tension model had larger values than that with the real tension model.

Simple and practical method for the simulations of two-component PFC models for binary colloidal crystals on curved surfaces

Herein, we propose a practical numerical method for solving two-component phase-field crystal model with vacancy effect on curved surfaces and perform various numerical simulations. In the present work, the curved surfaces are defined by the zero level-set of signed distance functions. By adopting the pseudo-Neumann boundary condition and closest-point type method, we can extend the computation into a three-dimensional narrow band domain which contains the curved surfaces. The spatial discretization is performed by using the standard finite difference method and the seven-point Laplacian operator can be used to replace the surface Laplace–Beltrami operator. Therefore, the numerical implementation is simple and efficient. The semi-implicit discretization with stabilization term is considered to update the solutions in time. By simulating the pattern formations on curved surfaces with various shapes, we validate the good performance of the proposed method.

A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials

A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.

A novel equivalent staggered-grid finite-difference scheme and its optimization strategy for variable-density acoustic wave modelling

Compared with the standard staggered-grid finite-difference (FD) methods, equivalent staggered-grid (ESG) ones can significantly reduce the computational memory for acoustic wave modelling in the variable-density media. To further enhance the simulation efficiency and accuracy, one way is to optimize the FD coefficients, another way is to design new FD stencils. In this paper, we propose a modified ESG (M-ESG) scheme which can significantly accelerate the wavefield simulation process while preserving or even improving the modelling accuracy. We calculate the FD coefficients by approximating the temporal and spatial derivatives simultaneously based on time–space domain (TS-D) dispersion relation of the discrete wave equation. Our M-ESG scheme in the TS-D can maintain basically the same accuracy as the conventional ESG (C-ESG) one when the FD coefficients are derived by the Taylor-series expansion (TE) approach. Note that the TS-D dispersion relation is nonlinear with respect to the FD coefficients of the C-ESG scheme, so it is difficult to obtain the optimized FD coefficients for the discrete wave equation. However, we can minimize the L2-norm error of the dispersion relation based on our M-ESG scheme to implement a linear FD coefficients optimization strategy, which is easy and efficient. Comparisons with TE- and optimization-based C-ESG schemes demonstrate the accuracy, stability, and efficiency superiorities of our TE- and optimization-based M-ESG ones.

A new finite difference algorithm for boundary value problems involving transmission conditions

Abstract The finite difference method (FDM) is used to find an approximate solution to ordinary and partial differential equations of various type using finite difference equations to approximate derivatives. The idea is to replace ordinary or partial derivatives appearing in the boundary-value problem by finite differences that approximate them. There is an extensive literature on this topic. But, as a rule, ordinary differential equations or partial differential equations were studied without an internal singular point and without corresponding transmission conditions. It is our main goal here to develop finite difference method to deal with an boundary value problem involving additional transmission conditions at the interior singular point. In this study, we have proposed a new modification of classical FDM for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transmission conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To demonstrate the efficiency and reliability of the proposed modification of FDM an illustrative example is solved b y this method. The obtained results are compared with those obtained by the standard FDM and by the analytical method. Corresponding graphical illustrations are also presented.

On the convergence and stability analysis of finite-difference methods for the fractional Newell-Whitehead-Segel equations

In this study, standard and non-standard finite-difference methods are proposed for numerical solutions of the time-spatial fractional generalized Newell-Whitehead-Segel equations describing the dynamical behavior near the bifurcation point of the Rayleigh-Benard convection of binary fluid mixtures. The numerical solutions have been found for high values of $p$ which shows the degree of nonlinear terms in the equations. The stability and convergence conditions of the obtained difference schemes are determined for each value of $p$. Errors of methods for various values of $p$ are given in tables. The compatibility of exact solutions and numerical solutions and the effectiveness of the methods are interpreted with the help of tables and graphics. It can be said that not only standard and non-standard finite-difference methods are feasible and effective methods to solve the given equation numerically but also useful in terms of computational cost and memory.

Computational Algorithms for the Analysis of Cancer Virotherapy Model

Cancer is a common term for many diseases that can affect any part of the body. In 2020, ten million people will die due to cancer. A worldwide leading cause of death is cancer by the World Health Organization (WHO) report. Interaction of cancer cells, viral therapy, and immune response are identified in this model. Mathematical and computational modeling is an effective tool to predict the dynamics of cancer virotherapy. The cell population is categorized into three parts like uninfected cells (x), infected cells (y), virus-free cells (v), and immune cells (z). The modeling of cancer-like diseases is based on the law of mass action (the rate of change of reacting substances is directly proportional to the product of interacting substances). Positivity, boundedness, equilibria, threshold analysis, are part of deterministic modeling. Later on, a numerical analysis is designed by using the standard and non-standard finite difference methods. The non-standard finite difference method is developed to study the long-term behavior of the cancer model. For its efficiency, a comparison of the methods is investigated.

A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ( nge 3). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.