Fourth-order Compact Difference Scheme Research Articles

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

<abstract><p>In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.</p></abstract>

Double reduction order method based conservative compact schemes for the Rosenau equation

In this paper, fourth-order compact difference schemes are derived, analyzed and tested at length for both one- and two-dimensional Rosenau equations under the spatial periodic boundary conditions on the basis of the double reduction order method and bilinear compact operator. We prove that these schemes satisfy mass and energy conservation law via the help of the energy method. In addition, the uniquely solvable, unconditional convergence and stability are all obtained with the convergence order four in space and order two in time under the L∞-norm. Several numerical examples are presented to support the theoretical results.

Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors

Understanding the connection between spatial patterns in population densities and ecological heterogeneity is significant to the understanding of population dynamics and the governance of species in a given domain. In this paper, the spatiotemporal complexity of prey–predator dynamics with fractional Laplacian derivative and migration is investigated. To provide good guidelines on the choice of parameters, the linear stability analysis of the models was investigated. The fractional Laplacian operator was defined in terms of the left- and right-handed Riemann–Liouville derivative which in turn were approximated by using the fourth-order compact difference scheme, and the resulting system of ODEs was advanced in time using the fourth-order exponential time-differencing Runge–Kutta method. In the simulation experiments, different Turing dynamics such as spots, stripes, and other chaotic patterns are observed. Overall, pattern formation in predator–prey models is useful for understanding the dynamics of ecological systems, predicting the long-term behavior of the system, and studying the impact of environmental factors on the dynamics of the system.

Local error estimates of the fourth-order compact difference scheme for a time-fractional diffusion-wave equation

This paper considers the numerical approximation for the time-fractional diffusion-wave equation with the order α∈(1,2), whose solution behaves a weak singularity at t=0. To construct the high-order scheme, the intermediate variable w=Dtα/2(u−tϕ˜) is introduced, then we can rewrite the original problem as a coupled system, where u is the solution of the time-fractional diffusion-wave equation and ϕ˜=ut(x,0). By using the L1 scheme on graded meshes in temporal direction and the compact difference scheme in spatial direction, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability and the local error estimate of the proposed scheme is given. Finally, numerical examples are presented to verify the accuracy and efficiency of the proposed method.

Fast TT-M fourth-order compact difference schemes for a two-dimensional space fractional Gray-Scott model

Fast TT-M fourth-order compact difference schemes for a two-dimensional space fractional Gray-Scott model

On the L$$^{\infty }$$ convergence of a novel fourth-order compact and conservative difference scheme for the generalized Rosenau-KdV-RLW equation

On the L$$^{\infty }$$ convergence of a novel fourth-order compact and conservative difference scheme for the generalized Rosenau-KdV-RLW equation

A physics-preserving pure streamfunction formulation and high-order compact solver with high-resolution for three-dimensional steady incompressible flows

In this paper, a pure streamfunction high-order compact (HOC) difference solver is proposed for three-dimensional (3D) steady incompressible flows. A physics-preserving pure streamfunction formulation is first introduced for the steady 3D incompressible Navier–Stokes (NS) equations without in-flow and out-flow boundary conditions, where the divergence of streamfunction ∇ · ψ is maintained in the convective and the vortex-stretching terms together in the nonlinear term of equations to reduce the physics-informed loss. Moreover, taking the streamfunction-vector components and their first-order partial derivatives as unknown variables, some fourth-order compact schemes are suggested for the partial derivatives that appear in the streamfunction formulation, and a high-resolution HOC scheme is introduced for approximating the pure third-order partial derivatives in the convective term. Meanwhile, a new HOC scheme is proposed for the first-type boundary conditions of the streamfunction. Finally, a fourth-order compact difference scheme and its algorithm are established for the 3D steady incompressible NS equations in the streamfunction form, subject to no in-flow and out-flow boundary conditions. Several numerical examples are carried out to validate and prove the accuracy, convergence, and efficiency of the proposed new method. Numerical results reveal that the proposed method not only can achieve fourth-order accuracy but also has excellent convergence, high-resolution, and low computational cost at higher Reynolds number.

Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers’ equation, from which L1-discretization formula is applied to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete L1 compact difference scheme on uniform meshes can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability of the proposed scheme are rigorously proved in the L∞-norm by the energy argument and mathematical induction. We also establish a temporal second-order compact difference scheme on graded time meshes for solving the problem with weak initial singularity. Finally, several numerical experiments are provided to test the accuracy of two numerical schemes and verify the theoretical analysis.

Explicit high-order compact difference method for solving nonlinear hyperbolic equations with three types of boundary conditions

In this paper, an explicit compact difference method is constructed to solve nonlinear hyperbolic equations with initial and three types of boundary conditions. Firstly, a nonlinear explicit fourth-order compact difference scheme is derived by truncation error residual correction and the fourth-order Padé approximation. The corresponding scheme for the linear hyperbolic equation is proven to be conditionally stable by the Fourier analysis method. Then, to improve the computational efficiency, the nonlinear scheme is linearized to obtain a linear explicit fourth-order compact difference scheme. Finally, numerical experiments are conducted to verify the stability and accuracy of the presented method.

A new high-order compact and conservative numerical scheme for the generalized symmetric regularized long wave equations

ABSTRACT In this paper, a new three-level implicit and fourth-order compact conservative difference scheme is developed for the generalized symmetric regularized long wave equations. The proposed scheme adopts a new time discretization method. The conservation of the new compact scheme is proved, and the discrete conservation laws are obtained. The priori estimation in the maximum norm is discussed by the discrete energy method and mathematical induction. Based on the priori estimation and Brouwer's fixed point theorem, the unique existence of the difference solution is proved. It is proved that the compact difference scheme is unconditionally convergent and stable in the maximum norm, and its convergence order is the fourth order in space and the second order in time. A decoupled and linearized iterative algorithm is proposed to calculate the nonlinear algebraic system generated by the compact scheme, and its convergence is proved. Several numerical experiments are performed to prove the efficiency and reliability of the proposed numerical scheme and to confirm the results obtained from the theoretical analysis.

Pointwise error analysis of the BDF3 compact finite difference scheme for viscous Burgers' equations

This paper formulates a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme based on a developed fourth-order operator for computing the approximate solution of the Burger's equation. The equation is one of the useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics and etc. This proposed approach approximates the solution of Burger's equation with the help of two main steps. In the first step, the temporal discretization is accomplished by virtue of the BDF3 approach. In the second step, a developed fourth-order operator and the classic compact difference formula combine with the method of order reduction are applied for spatial discretization, thereby constructing a fully-discrete scheme. The theoretical analysis is proved in detail by means of the discrete energy method. The proposed scheme is convergent with third order for time and fourth order for space. Numerical results are carried out to verify the validity and accuracy of the proposed method.

High-order compact finite difference schemes for solving the regularized long-wave equation

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ-weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media

In this article, we present an efficient numerical algorithm, which combines the fourth-order compact difference scheme (CDS) in space with the fast time two-mesh (TT-M) FBN-θ method, to solve the nonlinear distributed-order fractional Sobolev model appearing in porous media. We also construct the corrected CDS by adding the starting part to deal with the problem with nonsmooth solution and recovery the convergence rate. We derive optimal convergence results and prove the stability of the presented numerical algorithm. Finally, we implement numerical experiments by taking several examples with smooth and nonsmooth solutions to verify the correctness of the theoretical results, the effectiveness of the corrected algorithm and the computational efficiency of the fast algorithm.

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with in the discrete -norm is obtained. Here, and are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.

Numerical solution of the two-dimensional regularized long-wave equation with a conservative linearized high order finite difference scheme

In this article, a finite difference scheme is investigated to solve a two-dimensional regularized long-wave equation. The original equation is deformed into another formulation. The fourth order compact difference scheme and the fourth order Padé scheme are used for the treatment of the second and first derivatives of the deformation equation in space direction, and the fourth order backward difference formula is used for the discretization of the time derivative. The nonlinear term in the proposed scheme is linearized by Taylor series expansion method. The new scheme is shown to be of the fourth order accuracy in both temporal and spatial dimensions. The existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the new scheme is shown to be conservative and unconditionally stable. Finally, numerical experiments are provided to verify our theoretical analysis results of the new scheme.

A new linearized fourth-order conservative compact difference scheme for the SRLW equations

A new linearized fourth-order conservative compact difference scheme for the SRLW equations

A stabilized fully-discrete scheme for phase field crystal equation

A finite difference scheme is developed for solving the phase field crystal equation with the Dirichlet boundary condition. The second-order stabilized semi-implicit scheme is applied to discretize the time variable. The mass conservation, unique solvability of the semi-discrete scheme are proved. Then, the spatial discretization is attained by the fourth-order compact difference scheme. The unconditional energy stability and convergence analysis of the fully-discrete scheme are presented. In the implementation, a fast sine transform technique is utilized to reduce the computational cost. Several numerical examples are presented to verify the effectiveness of the proposed scheme.

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$