Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

In this paper,two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation.Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense,the new schemes are proved to per-fectly preserve the total energy in the discrete sense.By using the standard energy method and the cut-off function technique,the optimal error estimates of the numerical solutions are established,and the convergence rates are of O(h^(4)+τ^(2))with mesh-size h and time-step τ.In order to improve the computational efficiency,an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step.The convergence of the iterative algorithm is also rigorously analyzed.Several numerical results are carried out to test the error estimates and conservative properties.

Compact Difference Method for Time-Fractional Neutral Delay Nonlinear Fourth-Order Equation

In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal orders,or extending stencil widths,are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes.Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR)for nonlinear schemes.Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators.The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes,in terms of accuracy and resolution for smooth wave and vortex,especially for long-time simulations.Using optimized schemes increases the total CPU time by less than 4%.

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative

The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

The explicit compact difference scheme, proposed in Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD by Lin et al., published in Applied Mathematics and Mechanics （English Edition）, 2007, 28（7）, 943-953, has the same performance as the conventional finite difference schemes. It is just another expression of the conventional finite difference schemes. The proposed expression does not have the advantages of a compact difference scheme. Nonetheless, we can more easily obtain and implement compared with the conventional expression in which the coefficients can only be obtained by solving equations, especially for higher accurate schemes.

A Compact Explicit Difference Scheme of High Accuracy for Extended Boussinesq Equations

Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations. For time discretization, a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage, a cubic spline function is adopted at correcting stage, which made the time discretization accuracy up to fourth order; For spatial discretization, a three-point explicit compact difference scheme with arbitrary order accuracy is employed. The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme. The numerical results agree well with the experimental data. At the same time, the comparisons of the two numerical results between the present scheme and low accuracy difference method are made, which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations. As a valid sample, the wave propagation on the rectangular step is formulated by the present scheme, the modelled results are in better agreement with the experimental data than those of Kittitanasuan.

Overlapping Domain Decomposition Finite Difference Algorithm for Compact Difference Scheme of the Heat Conduction Equation

In this paper, a modified additive Schwarz finite difference algorithm is applied in the heat conduction equation of the compact difference scheme. The algorithm is on the basis of domain decomposition and the subspace correction. The basic train of thought is the introduction of the units function decomposition and reasonable distribution of the overlap of correction. The residual correction is conducted on each subspace while the computation is completely parallel. The theoretical analysis shows that this method is completely characterized by parallel.

High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids

In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.

A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

In this paper,a compact difference scheme is established for the heat equations with multi-point boundary value conditions.The truncation error of the difference scheme is O(τ2+h^4),where t and h are the temporal step size and the spatial step size.A prior estimate of the difference solution in a weighted norm is obtained.The unique solvability,stability and convergence of the difference scheme are proved by the energy method.The theoretical statements for the solution of the difference scheme are supported by numerical examples.

Optimization of a global seventh-order dissipative compact finite-difference scheme by a genetic algorithm

A global seventh-order dissipative compact finite-difference scheme is optimized in terms of time stability. The dissipative parameters appearing in the boundary closures are assumed to be different, resulting in an optimization problem with several parameters determined by applying a generic algorithm. The optimized schemes are analyzed carefully from the aspects of the eigenvalue distribution, the ε-pseudospectra, the short time behavior, and the Fourier analysis. Numerical experiments for the Euler equations are used to show the effectiveness of the final recommended scheme.

High Resolution Compact Finite Difference Schemes for Convection Dominated Problems

In this short article, the upwind and central compact finite difference schemes for spatial discretization of the first-order derivative are analyzed. Comparison of the schemes is provided and the best discretization scheme for convection dominated problems is suggested.

Prediction of Better Flow Control Parameters in MHD Flows Using a High Accuracy Finite Difference Scheme

We have successfully attempted to solve the equations of full-MHD model within the framework of Ψ- ωformulation with an objective to evaluate the performance of a new higher order scheme to predict better values of control parameters of the flow. In particular for MHD flows, magnetic field and electrical conductivity are the control parameters. In this work, the results from our efficient high order accurate scheme are compared with the results of second order method and significant discrepancies are noted in separation length, drag coefficient and mean Nusselt number. The governing Navier-Stokes equation is fully nonlinear due to its coupling with Maxwell’s equations. The momentum equation has several highly nonlinear body-force terms due to full-MHD model in cylindrical polar system. Our high accuracy results predict that a relatively lower magnetic field is sufficient to achieve full suppression of boundary layer and this is a favorable result for practical applications. The present computational scheme predicts that a drag-coefficient minimum can be achieved when β=0.4 which is much lower when compared to the value β=1 as given by second order method. For a special value of β=0.65, it is found that the heat transfer rate is independent of electrical conductivity of the fluid. From the numerical values of physical quantities, we establish that the order of accuracy of the computed numerical results is fourth order accurate by using the method of divided differences.

COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O（h^4） term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System

In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.

A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects,such as decreased accuracy and even numerical instability,of the entire computational method,especially for higher order methods.In this work,we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes(N-S)equations.In the pro-posed method,the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme.With such a feature,the nu-merical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme.Numerical examples show the effec-tiveness and accuracy of the present consistent compact high order scheme in L^(∞).Its application to two dimensional lid-driven cavity flow problem further exhibits that un-der the same condition,the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4^(th)order explicit scheme.The compact finite difference method equipped with the present consistent boundary technique im-proves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.

A High-Order Compact Scheme with Square-Conservativity

In order to improve the accuracy of forecasts of atmospheric and oceanic phenomena which possess a wide range of space and time scales, it is crucial to design the high-order and stable schemes. On the basis of the explicit square-conservative scheme, a high-order compact explicit square-conservative scheme is proposed in this paper. This scheme not only keeps the square-conservative characteristics, but also is of high accuracy. The numerical example shows that this scheme has less computing errors and better computational stability, and it could be considered to be tested and used in many atmospheric and oceanic problems.

Characteristic Analysis of Exponential Compact Higher Order Schemes for Convection-Diffusion Equations

This paper looks at the development of a class of Exponential Compact Higher Order (ECHO) schemes and attempts to comprehend their behaviour by introducing different combinations of discrete source function and its derivatives. The characteristic analysis is performed for one-dimensional schemes to understand the efficiency of the scheme and a similar analysis has been introduced for higher dimensional schemes. Finally, the developed schemes are used to solve several example problems and compared the error norms and rates of convergence.

A Compact Difference Scheme for Time-Space Fractional Nonlinear Diffusion-Wave Equations with Initial Singularity

In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solution is considered,which often generates a singular source and increases the difficulty of numerically solving the equation.The Crank-Nicolson technique,combined with the midpoint formula and the second-order convolution quadrature formula,is used for the time discretization.To increase the spatial accuracy,a fourth-order compact difference approximation,which is constructed by two compact difference operators,is adopted for spatial discretization.Then,the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space.Finally,numerical experiments are given to support our theoretical results.