High-order Compact Difference Scheme Research Articles

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Fast large-scale gravity modeling using a high-order compact cascadic multigrid strategy

A new high-order compact (HOC) cascadic multigrid (MG) strategy is developed for modeling large-scale complex gravity problems on nonuniform rectilinear grids. First, the governing 3D Poisson’s gravitational potential equation is discretized by a novel HOC difference scheme, which allows for Robin boundary conditions to further reduce the computational domain. Then, the resulting asymmetric linear system of equations is solved by a generalized extrapolation cascadic MG (gEXCMG) method augmented with a novel MG prolongation operator. The quartic interpolation and completed Richardson extrapolation assist this MG prolongation operator to produce a good initial guess for the biconjugate gradient stabilized smoother. Two synthetic and SEG/EAGE salt models are examined to verify the presented approach. Results indicate that our algorithm can efficiently and accurately produce the gravity signals for complex geologic models. Moreover, the comparisons with the state-of-the-art algebraic MG solvers demonstrate that our gEXCMG solver offers substantially better efficiency with significantly less memory consumption. Therefore, our newly developed method has the potential for serving as a forward engine in large-scale gravity inversions.

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

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>

Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

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>

A high-order compact difference scheme on graded mesh for time-fractional Burgers’ equation

A high-order compact difference scheme on graded mesh for time-fractional Burgers’ equation

Influence of the Mesh Topology on the Accuracy of Modelling Turbulent Natural and Excited Round Jets at Different Initial Turbulence Intensities

The paper is aimed at an assessment of the importance of the coordinate system (Cartesian vs. cylindrical) assumed for simulations of free-round jets. The research is performed by applying the large eddy simulation method with spatial discretisation based on high-order compact difference schemes. The results obtained for natural and excited jets at three different turbulence intensity levels, Ti=0.01%,0.1% and 1.0%, are compared. In the case of the natural jet, it is found that both instantaneous and time-averaged results are significantly dependent on the coordinate system only for the lowest Ti. In this case, in the Cartesian coordinate system, the errors introduced by an azimuthal non-uniformity of the mesh seem to have a larger impact on the solutions than the disturbances generated at the nozzle exit. The azimuthal non-uniformity of the mesh also has a substantial influence on the results of the modelling of the excited jets. In this case, the excitation is introduced as time-varying forcing, with the frequency corresponding to half of the preferred mode frequency and the amplitude equal to 5% of the jet velocity. Such an excitation leads to the formation of the so-called side-jets being revealed as inclined streams of fluid ejected outside the main jet stream. Primary attention is paid to the mechanism of the formation of the side-jets, their number and location. The results obtained on Cartesian meshes show that for very low turbulence intensity levels (Ti=0.01%), the number and direction of the side-jets are dependent on the non-uniform distribution of the mesh nodes along the azimuthal direction of the jet. On the other hand, when the cylindrical coordinate system is used, the number of the side-jets and their locations are random and dependent only on inlet parameters. It has been demonstrated that the mechanism of side-jet formation is the same in both coordinate systems; however, its random nature can only be predicted when the cylindrical coordinate system is used.

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.

Development of the LES-ADM approach for combustion modelling using high-order filters

The paper focuses on the development of an approximate deconvolution method (ADM) for large eddy simulation (LES) of turbulent reactive flow. We derive the effective filter as a product of the LES filter and the filter induced by spatial discretization performed using high-order compact difference schemes. We analyze the accuracy of the ADM depending on the effective filter definition in the case of modelling unsteady combustion phenomena (ignition and flame extinction) in a forced homogeneous isotropic turbulent field

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.

High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.

Application of High-Order Compact Difference Schemes for Solving Partial Differential Equations with High-Order Derivatives

In this paper, high-order compact-difference schemes involving a large number of mesh points in the computational stencils are used to numerically solve partial differential equations containing high-order derivatives. The test cases include a linear dispersive wave equation, the non-linear Korteweg–de Vries (KdV)-like equations, and the non-linear Kuramoto–Sivashinsky equation with known analytical solutions. It is shown that very high-order compact schemes, e.g., of 20th or 24th orders, cause a very fast drop in the L2 norm error, which in some cases reaches a machine precision already on relatively coarse computational meshes.

Numerical study of the flow around a hyperbolic cylinder at Reynolds number 3900

In this paper, a numerical computation method combining a high-order compact difference scheme and immersed boundary method is used to simulate the flow around a hyperbolic cylinder with different aspect ratios and eccentricity. It is found that compared to the circular cylinder, the vortex-separation point in the wake of the hyperbolic cylinder moves backward, and the corresponding drag coefficient decreases by 27.6%. More interestingly, the time-averaged drag coefficient of hyperbolic cylinder behaviors parabolic relationship with cross-section aspect ratio and hyperboloid eccentricity, respectively. As the aspect ratio decreases from 1 to 0.6, the flow drag parallel to the long-axis reduces dramatically to 42.9%. Especially, there is no obviously low pressure region behind the hyperbolic cylinder when r = 0.4, which leads to the drag coefficient is only 0.0249 and close to zero. Besides, the average drag coefficient diminishes in a parabolic trend with the eccentricity that has a monotonously positive correlation with two parameters, and an increase in eccentricity e by varying a or c will also reduce the drag, but the effect of parameter a is more obvious than parameter c. The results presented here are expected to suggest a strategy for drag reduction in flow around the cylinder and provide theoretical guidance for the design and optimization of hyperbolic structures in flowing fluid.

High-order compact difference schemes on wide computational stencils with a spectral-like accuracy

In this paper, we propose a generalized framework for the derivation of compact difference schemes for the approximation of an arbitrary order derivative. We formulate a methodology for calculating the approximation coefficients involving a large number of mesh points leading to very high-order formulas, e.g., 20th or 30th order. The main emphasis is on the first and second order derivative occurring in the Navier-Stokes equations governing fluid flow problems. Contrary to most studies on compact schemes in which the computational stencils employed in the discretization are narrow and consist of only a few mesh points K, in the presented paper we consider wide stencils that in practice are limited only by the overall number of the nodes in a computational grid. We analyze the features of such wide stencil based schemes in physical and spectral space. It is shown that the solutions obtained applying the schemes of the same order but involving different number of the mesh points characterize significantly different accuracy. This is presented by direct comparisons of the analytically derived formula for the leading terms of the truncation errors as well as in practical simulations. The test cases include a second order ordinary differential equation with variable coefficients, 1D Burgers equation and three typical benchmark problems used in Computational Fluid Dynamics (CFD), i.e., inviscid vortex advection, convected Taylor-Green vortices and a double jet flow. The obtained results are compared with available analytical solutions and the solutions obtained with the help of a pseudo-spectral method. We found that very high-order compact schemes reach the spectral accuracy characterized by the error decrease proportional to O(hK). In some cases, they show even higher accuracy, and moreover, reaching the limit of the machine precision they are substantially more resistant to the round-off errors.

High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

<abstract><p>In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.</p></abstract>

Numerical valuation of European and American options under Merton's model

In this paper, a new combination of time and spatial discretization is proposed for a partial integro-differential equation (PIDE) arising in the valuation of European options under Merton's model. We first present a high-order compact (HOC) difference scheme in space based on a uniform mesh to obtain a highly accurate result, and the discontinuous Galerkin (DG) finite element method in time is introduced that can deal with the loss of the time analyticity. A penalty method is proposed for a partial integro-differential complementarity problem arising in the valuation of the American put option. Numerical experiments are performed to verify the accuracy and efficiency of the proposed method.

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.