Fourth-order Compact Finite Difference Scheme Research Articles

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd s systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

Numerical Solution of Time-Fractional Schrödinger Equation by Using FDM

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme is utilized for discretizing the spatial term. The unconditional stability and convergence of the scheme in the maximum norm are proved. Finally, we verified the theoretical result with a numerical test.

Analysis of two conservative fourth-order compact finite difference schemes for the Klein-Gordon-Zakharov system in the subsonic limit regime

We propose two conservative fourth-order compact finite difference (CFD4C) schemes and give the rigorous error analysis for the Klein-Gordon-Zakharov system (KGZS) with ε∈(0,1] being a small parameter. In the case 0<ε≪1, i.e., the subsonic limit regime, the solution of KGZS propagates waves with wavelength O(ε) in time and O(1) in space, respectively, with amplitude at O(εα†) with α†=min⁡{α,β+1,2}, where α and β describe the incompatibility between the initial data of the KGZS and the limiting Klein-Gordon equation (KGE) as ε→0+ and satisfy α≥0,β+1≥0. Different from the standard KGZS, the oscillation in time becomes the main difficulty in constructing numerical schemes and giving the error analysis for KGZS in this regime. We prove that the schemes conserve the discrete energies. By defining a discrete ‘energy’, applying cut-off technique and considering the convergence rate of the KGZS to the limiting KGE, we obtain two independent error estimates with the bounds at O(h4/ε1−α⁎+τ2/ε3−α†) and O(h4+τ2+εα⁎) where α⁎=min⁡{1,α,1+β}, τ is time step and h is mesh size. Hence, we get uniformly accurate error estimates with the bounds at O(h4+τ2/(4−α†)) when α≥1 and β≥0, and O(h4α⁎+τ2α⁎/3) when 0<α<1 or −1<β<0. While for α=0 or β=−1, the meshing requirements of CFD4Cs are h=O(ε1/4) and τ=O(ε3/2) for 0<ε≪1. The compact schemes provide much better spatial resolution compared with the usual second order finite difference schemes. Our numerical results support energy-conserving properties and the results of the convergence analysis.

Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options

The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated.

An efficient extrapolation multigrid method based on a HOC scheme on nonuniform rectilinear grids for solving 3D anisotropic convection–diffusion problems

We develop an efficient multigrid method combined with a high-order compact (HOC) finite difference scheme on nonuniform rectilinear grids for solving 3D diagonal anisotropic convection–diffusion problems with boundary/interior layers. Firstly, we derive a fourth-order compact finite difference scheme to discretize the 3D anisotropic convection–diffusion equation on a rectilinear grid. Then, the resulting large-scale asymmetric linear system of equations is solved by a generalized extrapolation cascadic multigrid (gEXCMG) method based on two novel multigrid (MG) prolongation operators. The highlight of this paper is the application of the quintic Lagrange interpolation and the completed Richardson extrapolation in the design of the new MG prolongation operator on nonuniform rectilinear grids, which can produce a good initial guess (sixth-order approximation to the finite difference solution) for the SSOR-preconditioned biconjugate gradient stabilized (BiCGStab) smoother. In the end, numerical experiments show that the gEXCMG method combined with the HOC scheme can achieve fourth-order accuracy for 3D anisotropic convection–diffusion problems with few smoothing steps on the finest grid. Moreover, the proposed gEXCMG method can offer substantially better efficiency than the state-of-the-art algebraic MG solver, namely, aggregation-based algebraic multigrid (AGMG) method, for large linear systems arising from the discretization of second order elliptic PDEs.

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

A new three-level fourth-order compact finite difference scheme for the extended Fisher-Kolmogorov equation

In this paper, a new fourth-order compact finite difference scheme is proposed to solve the extended Fisher-Kolmogorov (EFK) equation. The scheme is three-level and implicit nonlinear. A new time discretization technique is adopted in the new scheme. The priori estimate and convergence of the numerical solution are obtained by using the discrete energy method, and the optimal error estimation in the maximum norm shows that the proposed numerical scheme has second-order accuracy in time and fourth-order accuracy in space. Finally, the theoretical analysis is supported by the numerical results of several examples.

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

In this paper, we propose and analyze a new fourth-order compact finite difference scheme for solving the quantum Zakharov system (QZS). The new scheme perfectly inherits the mass and energy conservation possessed by the QZS, while the energy preserved by the existing schemes expressed by two-level's solution at each time step. By using the energy method to analyze an equivalent form of the difference scheme, without any requirement on the grid ratio, we establish rigorously the optimal error estimate of the proposed scheme in the energy norm. The accuracy is shown to be fourth order in space and second order in time. Numerical examples are presented to verify the theoretical results and show the effectiveness of the proposed scheme.

A Fourth-Order Compact Finite Difference Scheme for Solving the Time Fractional Carbon Nanotubes Model.

In this work, we deal with unsteady magnetohydrodynamic allowed convection inflow of blood with a carbon nanotubes model; the single and multiwalled carbon nanotubes of human blood are used as a based fluid. Two numerical methods used to study this model are the weighted average finite difference method and the nonstandard compact finite difference method. The proportional Caputo hybrid operator has been used to fractionalize the proposed model. Stability analysis has been construed by a kind of John von Neumann stability analysis. Numerical results are presented in diverse graphs, which manifest that the method is successful in solving the proposed model.

Compact Finite Difference Scheme Combined with Richardson Extrapolation for Fisher’s Equation

In this study, the fourth-order compact finite difference scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation is presented. First, the derivative involving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term is linearized by the lagging method, and the derivative involving the temporal variable is discretized by the Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. The numerical results displayed in figures and tables show that the proposed method is efficient, accurate, and a good candidate for solving the 1D Fisher’s equation.

Transition to chaos in lid–driven square cavity flow* *Project supported by theNational Natural Science Foundation of China (Grant No. 12162001), the Natural Science Foundation of Ningxia (Grant No. 2019AAC03129), and the Construction Project of First-ClassDisciplines in Ningxia Higher Education (Grant No. NXYLXK2017B09).

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid–driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.

A compact Fourth-Order Implicit-Explicit Runge-Kutta Type Method for Solving Diffusive Lotka–Volterra System

This paper aims to developed a high-order and accurate method for the solution of one-dimensional Lotka-Volterra-diffusion with Numman boundary conditions. A fourth-order compact finite difference scheme for spatial part combined with implicit-explicit Runge Kutta scheme in temporal are proposed. Furthermore, boundary points are discretized by using a compact finite difference scheme in terms of fourth order accuracy. A key idea for proposed scheme is to take full advantage of method of line (MOL), this is consequently enabling us to use implicit-explicit Runge Kutta method, that are of fourth order in time. We constructed fourth order accuracy in both space and time and is unconditionally stable. This is consequently leading to a reduction in the computational cost of the scheme. Numerical experiments show that the combination of the compact finite difference with IMEX- RK methods give an accurate and reliable for solving the Lotka-Volterra-diffusion.

High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods

The main goal of this paper is to developed a high-order and accurate method for the solution of one-dimensional of generalized Burgers-Fisher with Numman boundary conditions. We combined between a fourth-order compact finite difference scheme for spatial part with diagonal implicit Runge Kutta scheme in temporal part. In addition, we discretized boundary points by using a compact finite difference scheme in terms of fourth order accuracy. This combine leads to ordinary differential equation which will take full advantage of method of line (MOL). Some numerical experiments presented to show that the combination give an accurate and reliable for solving the generalized Burgers-Fisher problems.

Analysis of a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation

In this paper, we propose and analyze a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise form of the proposed compact scheme into equivalent vector form and analyze its conservative and convergence properties. We prove that the new scheme conserves the total mass and energy in the discrete level and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of $$O(h^4 +\tau ^2)$$ in $$l^\infty $$ -norm, where h and $$\tau $$ are spatial and temporal steps, respectively. The techniques for error analysis include the energy method and the mathematical induction. The numerical experiments are carried out to confirm our theoretical analysis.

A fast forward algorithm for three-dimensional gravity gradient tensor using the compact difference schemes

We propose a fast and accurate 3D large-scale forward modeling method to compute the gravity gradient tensor, in terms of a fourth-order compact finite difference scheme and the extrapolation cascadic multigrid (EXCMG) method. Firstly, the 19-point fourth-order compact finite difference scheme with unequal mesh sizes in different coordinate directions is used to discretize the governing 3D Poisson equation. The resulted symmetric positive definite linear systems are solved by the EXCMG method. Then, the second-order derivatives (gravity gradient tensor) are calculated by solving a series of tridiagonal linear systems resulting from the fourth-order compact finite difference discretization. Finally, numerical result for a cubic model with positive density is used to verify the accuracy of our presented method. It shows that our method can give nearly fourth-order accurate approximations to gravitational potential and gravity gradient tensor. The accuracy is much higher and more efficient than traditional finite difference methods.

A Block–Interface Approach for High–Order Finite–Difference Simulations of Compressible Flows

The application of the high-order accurate schemes with multi-block domains is essential in problems with complex geometries. Primarily, accurate block-interface treatment is found to be of significant importance for precisely capturing discontinuities in such complex configurations. In the current study, a conservative and accurate multi-block strategy is proposed and implemented for a high-order compact finite-difference solver. For numerical discretization, the Beam-Warming linearization scheme is used and further extended for three-dimensional problems. Moreover, the fourth-order compact finite-difference scheme is employed for spatial discretization. The capability of the high-order multi-block approach is then evaluated for the onedimensional flow inside a Shubin nozzle, two-dimensional flow over a circular bump, and three-dimensional flow around a NACA 0012 airfoil. The results showed a reasonable agreement with the available exact solutions and simulation results in the literature. Further, the proposed block-interface treatment performed quite well in capturing shock waves, even in situations that the location of the shock coincides with block interfaces.

A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete $$L^{2}$$ -norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

A class of compact finite difference schemes for solving the 2D and 3D Burgers’ equations

In this paper, a class of two-level high-order compact finite difference implicit schemes are proposed for solving the Burgers’ equations. Firstly, based on the fourth-order compact finite difference schemes for spatial derivatives and the truncation error remainder correction method for temporal derivative, the high-order compact (HOC) difference method is introduced for solving the one-dimensional (1D) Burgers’ equation. At the same time, the stability of the scheme is analyzed by using the Fourier analysis method. Because only three grid points are involved in each time level. The Thomas algorithm can be directly used to solve the tridiagonal linear system. Then, this method is extended to solve the two-dimensional (2D) and three-dimensional (3D) coupled Burgers’ equations. Finally, numerical experiments are conducted to verify the accuracy and the reliability of the present schemes.

Optimal point-wise error estimate of two conservative fourth-order compact finite difference schemes for the nonlinear Dirac equation

In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O(h4+τ2) in l∞-norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations.