Linear Finite Difference Scheme Research Articles

A Linear Second-Order Finite Difference Scheme for the Allen–Cahn Equation with a General Mobility

A Linear Second-Order Finite Difference Scheme for the Allen–Cahn Equation with a General Mobility

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

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 linear finite difference scheme with error analysis designed to preserve the structure of the 2D Boussinesq paradigm equation

AbstractUse of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an ‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition.

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.

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

Half-sweep RSOR iteration with three-point linear rational finite difference scheme for solving first-order Fredholm integro-differential equations

Half-sweep RSOR iteration with three-point linear rational finite difference scheme for solving first-order Fredholm integro-differential equations

Half-sweep RSOR iteration with three-point linear rational finite difference scheme for solving first-order Fredholm integro-differential equations

Half-sweep RSOR iteration with three-point linear rational finite difference scheme for solving first-order Fredholm integro-differential equations

Numerical solutions of generalized Rosenau–KDV–RLW equation by using Haar wavelet collocation approach coupled with nonstandard finite difference scheme and quasilinearization

AbstractIn this article, we analyze and propose to compute the numerical solutions of a generalized Rosenau–KDV–RLW (Rosenau‐Korteweg De Vries‐Regularized Long Wave) equation based on the Haar wavelet (HW) collocation approach coupled with nonstandard finite difference (NSFD) scheme and quasilinearization. In the process of the numerical solution, the NSFD scheme is applied to discretize the first‐order time derivative, Haar wavelets are applied on spatial derivatives and the non‐linear term is taken care by quasilinearization technique. To discuss the efficiency of the method we compute error and error. We also use discrete mass and energy conservation to check the accuracy of the proposed methodology. The computed results have been compared with the existing methods, for example, three‐level average implicit finite difference technique, B‐spline collocation, three‐level linear conservative implicit finite difference scheme and conservative fourth‐order stable finite difference scheme.

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.

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper, we develop two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain nonlinear high order derivatives, and possibly peakon solutions or shock waves. By introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic system, and the \mdp equation as a first order system. Then we choose a linear finite difference scheme with suitable order of accuracy for the auxiliary variable(s), and two finite difference WENO schemes with unequal-sized sub-stencils for the primal variable. One WENO scheme uses one large stencil and several smaller stencils, and the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil, both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights. Another advantage is that the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

Conservative compact difference scheme for the generalized Korteweg-de Vries equation

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

Linear and fully decoupled scheme for a hydrodynamics coupled phase-field surfactant system based on a multiple auxiliary variables approach

We propose a linear, fully decoupled, and energy stable finite difference scheme for solving a phase-field surfactant fluid system. Inspired by the idea of multiple scalar auxiliary variables (MSAV) approach, two scalar auxiliary variables are used to transform the original governing equations into their equivalent forms. Based on the equivalent system, a highly efficient scheme can be developed. In one time cycle, the proposed algorithm can be efficiently performed, i.e., the surfactant ψ is explicitly updated, then the phase-field function ϕ, velocity field u, and pressure field p can be computed by solving linear systems with constant coefficients. The energy dissipation law for a modified energy can be estimated by using the proposed method. Various computational simulations confirm that the proposed method is not only accurate and energy stable but also works well for simulating surfactant-laden droplet dynamics.

A linear finite-difference scheme for approximating randers distances on cartesian grids

Randers distances are an asymmetric generalization of Riemannian distances, and arise in optimal control problems subject to a drift term, among other applications. We show that Randers eikonal equation can be approximated by a logarithmic transformation of an anisotropic second order linear equation, generalizing Varadhan’s formula for Riemannian manifolds. Based on this observation, we establish the convergence of a numerical method for computing Randers distances, from point sources or from a domain’s boundary, on Cartesian grids of dimension 2 and 3, which is consistent at order 2/3, and uses tools from low-dimensional algorithmic geometry for best efficiency. We also propose a numerical method for optimal transport problems whose cost is a Randers distance, exploiting the linear structure of our discretization and generalizing previous works in the Riemannian case. Numerical experiments illustrate our results.

A Linear Finite Difference Scheme for the Two-Dimensional Nonlinear Schrödinger Equation with Fractional Laplacian

A Linear Finite Difference Scheme for the Two-Dimensional Nonlinear Schrödinger Equation with Fractional Laplacian

A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

In this paper, we propose a new linear second-order finite difference scheme for Allen–Cahn equations. We use a modified Leap-Frog finite difference scheme with stabilized term and a central finite difference scheme for temporal and spatial discretization respectively. It is shown that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Based on the maximum stability, the maximum-norm error is analyzed. Moreover, we can see that the proposed scheme is unconditionally energy-stable. Finally, a numerical experiment is performed to verify the theoretical results.

Numerical study on spatial scale characteristics of sound scattering by a static isentropic vortex

The scattering of acoustic waves by a vortex is a fundamental problem of the acoustic waves propagation in complex flow field, which plays an important role in academic research and engineering application for sound source localization, acoustic target recognition and detection, the far field noise prediction, such as aircraft wake vortex identification, detection and ranging, acoustic target forecasting in turbulent shear flow, acoustic measurement and sound source localization in wind tunnel test, etc. The nonlinear scattering phenomenon occurs when acoustic wave passes through the vortex, which is mainly related to the length-scale ratio between the wavelength of acoustic wave and the core radius of the vortex. In this paper, a plane acoustic wave passing through a stationary isentropic vortex is numerically simulated by solving a two-dimensional compressible, unsteady Euler equation. A sixth-order linear compact finite difference scheme is employed for spatial discretization. Time integration is performed by a four-stage fourth-order Runge-Kutta method. The eighth-order spatial compact filter scheme is adopted to suppress high frequency errors. At the far field boundaries, buffer layer is applied to handle the outgoing acoustic wave. Under the matching condition, the accuracy of the numerical results is verified by comparing with the previous direct numerical simulation results. The acoustic scattering cross-section method is introduced to analyze the effects of different length-scale ratio on the acoustic pulsation pressure, acoustic scattering effective sound pressure and acoustic scattering energy. Scattering occurs when sound waves pass through the vortex, the acoustic field in front of the vortex is basically unaffected, and the acoustic wave front remains intact. A “vacuum” region is formed slightly below the acoustic field directly behind the vortex, and two primary interference bands and several secondary interference bands are formed on the upper and lower sides of the vortex. As the length-scale ratio increases, the sound scattering decreases and the influence of the vortex flow field on the acoustic field gradually weakens. The influence region of effective sound pressure of acoustic scattering is mainly concentrated behind the vortex. With the increase of the length scale ratio, the influence gradually increases and extends to the upstream, and then the influence region gradually decreases to the vicinity of the vortex. When the length scale ratio is greater than or equal to 6, the location of the maximum effective sound pressure of sound scattering jumps from the upper right to the lower right of the vortex. The influence of acoustic wave wavelength change on the acoustic scattering energy can be divided into three parts. With the increase of the length scale ratio, the maximum sound scattering energy presents four different stages.

Analytical and numerical techniques for initial‐boundary value problems of Kolmogorov–Petrovsky–Piskunov equation

AbstractIn this work, we are interested in finding exact and numerical solutions of well‐known Kolmogorov–Petrovsky–Piskunov (KPP) equation. Here, we introduce modified extended tanh method for finding the exact solutions of KPP equation and an average linear finite difference scheme for numerical investigations. It has been observed that the numerical scheme so employed is stable and accurate enough to produce stable results.

FEM-IDS for a second order strongly damped wave equation with memory

A linear triangular finite element method-implicit difference scheme (FEM-IDS) for the solution of second order strongly damped wave equation (SDWE) with memory on domain with interfaces is proposed. Sufficient conditions that guarantee the existence of a unique solution are given. The finite element discretization is such that the arbitrary (but smooth) interface is first approximated by a polygon with a boundary whose vertices all lie on the interface. With the interface being at σ-distance from the approximate interface, linear interpolation operator incorporating the influence of σ isproved. This together with Ritz–Volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme. The scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method.

ON Numerical Blow-Up Solutions of Semilinear Heat Equations

This paper is concerned with the numerical blow-up solutions of semi-linear heat equations, where the nonlinear terms are of power type functions, with zero Dirichlet boundary conditions. We use explicit linear and implicit Euler finite difference schemes with a special time-steps formula to compute the blow-up solutions, and to estimate the blow-up times for three numerical experiments. Moreover, we calculate the error bounds and the numerical order of convergence arise from using these methods. Finally, we carry out the numerical simulations to the discrete graphs obtained from using these methods to support the numerical results and to confirm some known blow-up properties for the studied problems.