Second-order Finite Difference Scheme Research Articles

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

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.

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal L∞-error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results.

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

A second-order [formula omitted]-[formula omitted] difference scheme for the nonlinear time–space fractional Schrödinger equation

In this paper, we mainly consider the numerical approximation for the nonlinear time–space fractional Schrödinger equation with the Dirichlet boundary condition. A linearized second-order finite difference scheme is presented by adopting the L2-1σ formula to approximate the Caputo fractional derivative and the second-order weighted and shifted Grünwald difference formula to approximate the Riesz fractional derivative. To our knowledge, obtaining numerical solutions in the numerical scheme with the nonlinear term is very expensive. In order to reduce the computational cost and memory, the nonlinear term is handled by the local extrapolation method, which is more efficient than implicit processing and without reduced the convergence accuracy. Meanwhile, the numerical scheme of the coupled nonlinear time–space fractional Schrödinger system is also presented. The optimal error estimate of the numerical solution with convergence order O(τ2+h2) is strictly proven. Finally, a series of numerical experiments are presented to confirm the accuracy and effectiveness of the proposed numerical schemes.

Hyers–Ulam Stability of a Second-Order Finite Difference Scheme Using a Diamond-$$\alpha $$ Difference Operator

Hyers–Ulam Stability of a Second-Order Finite Difference Scheme Using a Diamond-$$\alpha $$ Difference Operator

Model and numerical method for soliton propagation through thermal medium based on nonlinear Schrödinger and heat transfer equations

We present a mathematical model and numerical method for simulating soliton propagation through thermal medium. The model is developed based on the nonlinear Schrödinger (NLS) equation coupled with a heat transfer equation. The well-posedness of the model and stability of the solution are analyzed. We then develop a second-order accurate finite difference scheme for solving this problem and prove the numerical solution to be bounded in l∞ norm. Furthermore, the scheme is proved to be second-order convergent in l2 norm. Numerical examples are given to verify the model and test the accuracy of the numerical scheme for simulating soliton propagations through the thermal medium.

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensions

A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensions

Preconditioned successive over relaxation iterative method via semi-approximate approach for Burgers’ equation

<span lang="EN-US">This paper proposes the combination of a preconditioner applied with successive over relaxation (SOR) iterative method for solving a sparse and huge scale linear system (LS) in which its coefficient matrix is a tridiagonal matrix. The purpose for applying the preconditioner is to enhance the convergence rate of SOR iterative method. Hence, in order to examine the feasibility of the proposed iterative method which is preconditioner SOR (PSOR) iterative method, first we need to derive the approximation equation of one-dimensional (1D) Burgers’ equation through the discretization process in which the second-order implicit finite difference (SIFD) scheme together with semi-approximate (SA) approach have been applied to the proposed problem. Then, the generated LS is modified into preconditioned linear system (PLS) to construct the formulation of PSOR iterative method. Furthemore, to analyze the feasibility of PSOR iterative method compared with other point iterative methods, three examples of 1D Burgers’ equation are considered. In conclusion, the PSOR iterative method is superior than PGS iterative method. The simulation results showed that our proposed iterative method has low iteration numbers and execution time.</span>

Controlled-source electromagnetic modeling using a high-order finite-difference time-domain method on a nonuniform grid

Simulation of 3D low-frequency electromagnetic (EM) fields propagating in the earth is computationally expensive. We have developed a fictitious wave-domain high-order finite-difference time-domain (FDTD) modeling method on nonuniform grids to compute frequency-domain 3D controlled-source EM data. The method overcomes the inconsistency issue widely present in the conventional second-order staggered-grid finite-difference scheme over a nonuniform grid, achieving high accuracy with an arbitrarily high-order scheme. The finite-difference coefficients adaptive to the node spacings can be accurately computed by inverting a Vandermonde matrix system using an efficient algorithm. A generic stability condition applicable to nonuniform grids is established, revealing the dependence of the time step and these finite-difference coefficients. A recursion scheme using fixed-point iterations is designed to determine the stretching factor to generate the optimal nonuniform grid. The grid stretching in our method reduces the number of grid points required in the discretization, making it more efficient than the standard high-order FDTD with a densely sampled uniform grid. Instead of stretching in vertical and horizontal directions, better accuracy of our method is observed when the grid is stretched along the depth without horizontal stretching. The efficiency and accuracy of our method are demonstrated by numerical examples.

An SBP-SAT FDTD Subgridding Method Using Staggered Yee’s Grids Without Modifying Field Components for TM Analysis

A summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method is proposed to model geometrically fine structures in this article. Compared with other works, the proposed SBP-SAT FDTD method uses the staggered Yee’s grid without adding or modifying any field components through field extrapolation on the boundaries to make the discrete operators satisfy the SBP property. The accuracy of extrapolation keeps consistency with that of the second-order finite-difference scheme near the boundaries. In addition, the SATs are used to weakly enforce the tangential boundary conditions between multiple mesh blocks with different mesh sizes. With carefully designed interpolation matrices and selected free parameters of the SATs, no dissipation occurs in the whole computational domain. Therefore, its long-time stability is theoretically guaranteed. Four numerical examples are carried out to validate its effectiveness. Results show that the proposed SBP-SAT FDTD subgridding method is stable, accurate, efficient, and easy to implement based on the existing FDTD codes with only a few modifications.

High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures.

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

Temporal second-order finite difference schemes for variable-order time-fractional generalized Oldroyd-B fluid model

In this paper, we study the variable-order generalized time fractional Oldroyd-B fluid model, use the reduced order method and the L2-1? method to establish the differential format with second-order accuracy, prove the stability and convergence of the format, and give numerical examples to illustrate the effectiveness of the differential format.

Study on the Improved Method for Calculating Traveltime and Raypath of Multistage Fast Marching Method

The traditional Fast Marching Method (FMM) based on the finite-difference scheme can solve the traveltime of first arrivals; however, the accuracy and efficiency of FMM are usually affected by the finite-difference schemes and grid size. The Vidale finite-difference scheme and double-grid technology are adopted to replace the traditional first-order and second-order finite-difference schemes in this paper to improve the computation accuracy and efficiency. The traditional FMM does not provide the corresponding raypath calculation methods, and in view of the interoperability of FMM and the linear travel time interpolation (LTI) method, we introduce the linear interpolation method into FMM ray tracing to compute the raypath and take into consideration the secondary source located inside the grid cell to improve the accuracy and stability of the raypath calculation. With these measures and the application of the multistage approach, we successfully completed the improved Multistage FMM (MFMM) ray tracing, which can track first arrivals and any type of primary and multiple reflection waves. Through the theoretical and actual field model tests, the computation accuracy and efficiency of the improved MFMM are proven to be higher than that under traditional first-order and second-order finite-difference schemes, the correctness and effectiveness of the interpolation method for raypath calculation are verified, and the improved MFMM has demonstrated good adaptability and stability for complex models. The improvements for the MFMM in this paper are successfully applied in two-dimensional cases and need to be extended to three-dimensional situations.

Ulam–Hyers Stability of Second-Order Convergent Finite Difference Scheme for First- and Second-Order Nonhomogeneous Linear Differential Equations with Constant Coefficients

Ulam–Hyers Stability of Second-Order Convergent Finite Difference Scheme for First- and Second-Order Nonhomogeneous Linear Differential Equations with Constant Coefficients

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

A phase-field method for elastic mechanics with large deformation

Boundary tracking is a challenging numerical problem for elastic mechanics problems with large deformation. In particular, when the boundary conditions rely on the boundary shape, iteration is required between tracking boundaries and solving the elasticity equations. Stability of this iteration brings additional difficulty for designing efficient numerical methods. In this work, we introduce phase-field variables to represent different elastic domains and track the movement of boundaries. Furthermore, boundary conditions relying on the boundary shape can also be easily applied with the phase-field variables. Second-order compact finite difference schemes are used to solve the coupled system on a square-mesh staggered grid. Our numerical results show that the phase-field method can be conveniently used to solve complex elastic mechanics problems accurately.

A Numerical Method for a Heat Conduction Model in a Double-Pane Window

In this article, we propose a one-dimensional heat conduction model for a double-pane window with a temperature-jump boundary condition and a thermal lagging interfacial effect condition between layers. We construct a second-order accurate finite difference scheme to solve the heat conduction problem. The designed scheme is mainly based on approximations satisfying the facts that all inner grid points has second-order temporal and spatial truncation errors, while at the boundary points and at inter-facial points has second-order temporal truncation error and first-order spatial truncation error, respectively. We prove that the finite difference scheme introduced is unconditionally stable, convergent, and has a rate of convergence two in space and time for the L∞-norm. Moreover, we give a numerical example to confirm our theoretical results.

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results.