Compact Difference Method Research Articles

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

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.

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations

In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Accelerated schemes of compact difference methods for space‐fractional sine‐Gordon equations with distributed delay

AbstractIn this paper, for quickly solving one‐ and two‐dimensional space‐fractional sine‐Gordon equations with distributed delay, we suggest several accelerated schemes of direct compact difference (DCD) methods. For one‐dimensional (1D) problems, with a function transformation, we construct an indirect compact difference (ICD) method, which requires less calculation cost than the corresponding DCD method, and prove under the appropriate conditions that ICD method has second‐order (resp. forth‐order) calculation accuracy in time (resp. space). By extending the argument for 1D case, we further obtain an ICD method for solving two‐dimensional (2D) problems and derive the similar convergence result. For ICD and DCD methods of 2D problems, we also give their alternative direction implicit (ADI) schemes. Moreover, for the fast implementations of ICD method of 1D problems and indirect ADI method of 2D problems, we further present their acceleration strategies. Finally, with a series of numerical experiments, the findings in this paper are further confirmed.

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

An Efficient Compact Difference Method for the Fourth-order Nonlocal Subdiffusion Problem

An Efficient Compact Difference Method for the Fourth-order Nonlocal Subdiffusion Problem

A high-order and efficient numerical technique for the nonlocal neutron diffusion equation representing neutron transport in a nuclear reactor

In this paper, a high-order and efficient numerical technique is constructed to solve nonlocal neutron diffusion equation with delayed neutrons representing neutron transport in a nuclear reactor. The method is based on approximating the temporal derivative by L1-2 technique, in combination with a space discretization by using compact difference method. The convergence of this method are studied using energy analysis and Cholesky decomposition. The error convergence order is shown to be O(k3−2α+h4), where α is the order of fractional derivative, k and h represent the parameters for the time and space meshes, respectively. Further, two numerical experiments are presented to validate the sharpness of our theoretical error bounds.

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black–Scholes model

The Black–Scholes (B–S) equation has been recently extended as a kind of tempered time-fractional B–S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B–S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.

High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

A high-order compact difference method and its Richardson extrapolation for semi-linear reaction–diffusion equations with piecewise continuous argument in diffusion term

In this paper, for the initial–boundary value problems of semi-linear reaction–diffusion equations with piecewise continuous argument in spatial derivative, we suggest Crank–Nicolson method, high-order compact difference (HOCD) method and HOCD-based Richardson extrapolation (RHOCD) method. Under the appropriate conditions, it is proved that HOCD (resp. RHOCD) method has the computational accuracy O(τ2+h4) (resp. O(τ4+h4)). This shows that RHOCD method improves the calculation accuracy of HOCD method in temporal direction. Moreover, we also analyze the stability of HOCD method and thus derive a global stability criterion of this method. Finally, with a series of numerical experiments, we further confirm the computational effectiveness and theoretical accuracy of the concerned methods.

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.

Impact of a discretisation method and chemical kinetics on the accuracy of simulation of a lifted hydrogen flame

This paper presents an analysis of numerical and modelling issues based on a simulation of nitrogen-diluted hydrogen lifted flame evolving in a hot co-flow. We apply the large-eddy simulations (LES) method with the so-called ‘no combustion model’ and concentrate on the impact of chemical mechanisms and various discretisation schemes on the obtained results. The main attention is put to the latter issue in which we analyse to what extent a discretisation method of the convective terms of the scalar equations for the species and enthalpy affects the solutions. We consider eight commonly known total variation diminishing (TVD) schemes and three upwind schemes of the second order. The remaining terms in the scalar equations and all the terms in the Navier–Stokes equations are discretised applying the sixth-order compact difference method. Such an approach makes the discretisation errors of the convective terms the main factor affecting the solutions from the numerical point of view. Prior to the main analysis, the differences caused by the use of various TVD or upwind schemes are highlighted based on a single scalar transport equation with a known analytical solution. The results obtained for the flame are compared to experimental data taken from the literature. It is shown that the differences due to the application of a particular discretisation method are of similar magnitude as the differences between the simulation results and experimental data. Moreover, analysis of the impact of the chemical mechanism showed that observed differences are comparable to these originating from the use of different discretisation methods.

Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation

<abstract><p>In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.</p></abstract>

High-order compact difference methods for solving two-dimensional nonlinear wave equations

<abstract><p>Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.</p></abstract>

Linearized compact difference schemes applied to nonlinear variable coefficient parabolic equations with distributed delay

AbstractIn this article, combined with the compound trapezoidal formula, two linearized compact difference methods are presented for solving nonlinear variable coefficient parabolic equations with distributed delay. It is proved under some appropriate conditions that these linearized compact schemes are uniquely solvable and convergent of order two in time and order four in space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

AbstractThis paper deals with the numerical computation and analysis for nonlinear Sobolev equations with distributed delay. We present linearized compact difference methods for solving one‐ and two‐dimensional cases, respectively. The solvability, convergence and stability of the methods are derived and it is shown under some suitable conditions that the methods have second‐order accuracy in time and fourth‐order accuracy in space. Moreover, with several numerical examples, the theoretical accuracy and computational effectiveness of the proposed methods are confirmed.