Compact Finite Difference Scheme Research Articles

A Compact High-Order Finite-Difference Method with Optimized Coefficients for 2D Acoustic Wave Equation

High-precision finite difference (FD) wavefield simulation is one of the key steps for the successful implementation of full-waveform inversion and reverse time migration. Most explicit FD schemes for solving seismic wave equations are not compact, which leads to difficulty and low efficiency in boundary condition treatment. Firstly, we review a family of tridiagonal compact FD (CFD) schemes of various orders and derive the corresponding optimization schemes by minimizing the error between the true and numerical wavenumber. Then, the optimized CFD (OCFD) schemes and a second-order central FD scheme are used to approximate the spatial and temporal derivatives of the 2D acoustic wave equation, respectively. The accuracy curves display that the CFD schemes are superior to the central FD schemes of the same order, and the OCFD schemes outperform the CFD schemes in certain wavenumber ranges. The dispersion analysis and a homogeneous model test indicate that increasing the upper limit of the integral function helps to reduce the spatial error but is not conducive to ensuring temporal accuracy. Furthermore, we examine the accuracy of the OCFD schemes in the wavefield modeling of complex structures using a Marmousi model. The results demonstrate that the OCFD4 schemes are capable of providing a more accurate wavefield than the CFD4 scheme when the upper limit of the integral function is 0.5π and 0.75π.

A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.•An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.•An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.•The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

An implicit fully discrete compact finite difference scheme for time fractional diffusion-wave equation

<abstract><p>In this paper, an implicit compact finite difference (CFD) scheme was constructed to get the numerical solution for time fractional diffusion-wave equation (TFDWE), in which the time fractional derivative was denoted by Caputo-Fabrizio (C-F) sense. We proved that the full discrete scheme is unconditionally stable. We also proved that the rate of convergence in time is near to $ O(\tau^{2}) $ and the rate of convergence in space is near to $ O(h^{4}) $. Test problem was considered for regular domain with uniform points to validate the efficiency and accuracy of the method. The numerical results can support the theoretical claims.</p></abstract>

A general conservative eighth-order compact finite difference scheme for the coupled Schrödinger-KdV equations

<abstract><p>In this paper, we present a general conservative eighth-order compact finite difference scheme for solving the coupled Schrödinger-KdV equations numerically. The proposed scheme is decoupled and preserves several physical invariants in discrete sense. The matrices obtained in the eighth-order compact scheme are all circulant symmetric positive definite so that it can be used to solve other similar equations. Numerical experiments on model problems show the better performance of the scheme compared with other numerical schemes.</p></abstract>

Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.

Effect of arc‐shaped vertical control plate on heat and mass transfer in uniform flow past an isothermally heated circular cylinder

AbstractThe current work investigates the effect of an arc‐shaped vertical control plate on the heat and mass transfer in uniform flow past an isothermally heated circular cylinder. The control plate is positioned downstream at various distances from the circular cylinder's surface. The governing equations are discretized by the higher order compact (HOC) finite difference scheme, and then this system of discretized equations is solved by using a bi‐conjugate gradient stabilized iterative method for Prandtl number , Reynolds number . To investigate the effect of an arc‐shaped control plate on heat and mass transfer, we consider a range of nondimensional distances between the circular cylinder and the control plate, , where is the control plate's distance and is the cylinder's radius. The exact timing and location of the bifurcation points are calculated by using topological aspect‐based structural bifurcation analysis. Significant effects of various locations of the control plate on periodic wake and heat transfer are observed. It is found that the increasing distance of the control plate from the cylinder delays the occurrence of the structural bifurcation and shifts the bifurcation points upwards in the upper half and downwards in the lower half of the cylinder. Our study shows that the specific location of the control plate can fully suppress the vortex shedding. Time‐averaged total Nusselt number can be drastically reduced by increasing the distance between the control plate and the cylinder. With proper positioning, the vertical control plate can lessen the time‐averaged drag force by up to when compared to a cylinder without a control plate. Overall, this work presents many new phenomena that have not been reported before.

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.

Non-Local Mean Denoising Algorithm Based on Fractional Compact Finite Difference Scheme Effectively Reduces Speckle Noise in Optical Coherence Tomography Images.

Optical coherence tomography (OCT) is used in various fields such, as medical diagnosis and material inspection, as a non-invasive and high-resolution optical imaging modality. However, an OCT image is damaged by speckle noise during its generation, thus reducing the image quality. To address this problem, a non-local means (NLM) algorithm based on the fractional compact finite difference scheme (FCFDS) is proposed to remove the speckle noise in OCT images. FCFDS uses more local pixel information when compared to integer-order difference operators. The FCFDS operator is introduced into the NLM algorithm to construct a high-precision weight calculation so that the proposed algorithm can effectively reduce the speckle noise in the OCT images. Experiments on simulations and real OCT images show that the proposed method is comparable to other state-of-the-art despeckling methods and can substantially reduce noise and preserve image details such as edges and structures. Speckle noise removal can further promote the application of the proposed algorithm in medical diagnosis and industrial detection, as it has key research value.

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.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

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.

Efficacy of line-based explicit and compact high-order finite difference schemes for hybrid unstructured grids

This study enables the use of explicit and compact high-order finite-difference schemes with a line-based solver to solve conservation laws on unstructured grids. A quadrilateral subdivision process is used to identify unique line structures (also known as Hamiltonian loops) before formulating stencil-based discretizations. To demonstrate the methodology for canonical flows represented by the Navier–Stokes equations, up to sixth-order spatial discretization and a maximum of tenth-order low-pass filters are implemented along with the Hamiltonian loops. The filter restores the benefits of the high-order approach even in the presence of abrupt grid discontinuities that cause abrupt changes in loop curvature. Isentropic vortex convection, lid-driven cavity, double periodic shear layer, and inviscid and viscous flow past a cylinder and NACA 0012 airfoil are all used to demonstrate the formulation. The ability of some schemes, particularly the fourth-order explicit, to nearly achieve the formal order of accuracy and successfully predict the flow following available results in the literature is one of the key findings. Compact schemes have been found to be more sensitive to loop curvature than explicit schemes.

High-performance computing of structure-preserving algorithm for the coupled BBM system formulated by weighted compact difference operators

A compact difference operator is a valuable technique which has been used to develop numerical schemes capable of solving various types of differential equations. Advantages of compact procedures include obtaining high accuracy with a relatively small number of grid points. In this paper, we introduce weighted parameters in standard compact difference operators. Therefore, applications of both high-order compact finite difference operators for spatial derivatives and the Crank–Nicolson/Adams–Bashforth method for temporal derivatives have been used to solve coupled BBM equations. As the consequence system of equations is uncoupled during computing, it requires less CPU processing time. In addition, when producing numerical solutions, the system equations obtained at each time step are constant-coefficient matrices, gaining a remarkable number of computational advantages. In addition, deriving an a priori estimate of numerical solutions obtains convergence and stability analyses. Simultaneously, the algorithm is globally structure-preserving as to the negligence of nonphysical behavior. Numerical results demonstrate that the proposed weighted compact finite difference scheme can improve the computational efficiency of standard second-order methods. Additionally, appropriately weighted parameters can refine numerical accuracy. Lastly, an investigation of the relevant properties of numerically computed solutions is conducted, including a finite time blow-up of a head-on collision and a two-way wave propagation.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)].

A phase-field method for elastic mechanics with large deformation

Boundary tracking is a challenging numerical problem for elastic mechanics problems with large deformation. In particular, when the boundary conditions rely on the boundary shape, iteration is required between tracking boundaries and solving the elasticity equations. Stability of this iteration brings additional difficulty for designing efficient numerical methods. In this work, we introduce phase-field variables to represent different elastic domains and track the movement of boundaries. Furthermore, boundary conditions relying on the boundary shape can also be easily applied with the phase-field variables. Second-order compact finite difference schemes are used to solve the coupled system on a square-mesh staggered grid. Our numerical results show that the phase-field method can be conveniently used to solve complex elastic mechanics problems accurately.

An explicit high-order compact finite difference scheme for the three-dimensional acoustic wave equation with variable speed of sound

In this paper, the correction for the remainder of the truncation error of the second-order central difference scheme is employed to discretize temporal derivative, while the fourth-order Padé schemes are directly used to compute spatial derivatives, an explicit high-order compact finite difference scheme to solve the three-dimensional acoustic wave equation with variable speed of sound is proposed. This new scheme has the fourth-order accuracy in both temporal and spatial directions. It has high computational efficiency since the Thomas algorithm is employed to solve three tridiagonal linear systems formed by the Padé schemes on the (n)th time step and then an explicit time advancement process is conducted for the th time step. The stability and convergence condition of the proposed scheme are proved. We extend the proposed method to solve problems with nonlinear source terms and systems. Numerical experiments are conducted to demonstrate theoretical analysis results of the proposed scheme.

Fast compact finite difference schemes on graded meshes for fourth-order multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions

In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes.

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

Magneto-thermogravitational convection for hybrid nanofluid in a novel shaped enclosure

The proposed work exhibits a numerical simulation of steady thermogravitational convection of Cu–Al2O3(50%−50%) water hybrid magneto-nanofluid in a novel shaped enclosure regulated by magnetic field dependent (MFD) viscosity in presence of flush mounted variable heaters. Various cases (Case-I to Case-IV) have been considered depending on the location of the heaters on the bottom and left boundaries. Pertaining to this analysis, the streamfunction (ψ)- vorticity (ζ) form including the energy transport equation of coupled partial differential equations are evaluated with the help of an efficient higher order compact finite difference scheme. Results are exhibited in terms of streamlines, isotherms and Nusselt numbers to describe the transport phenomena obtained for different physical parametric values to this simulation. The influence of the physical parameters such as, Rayleigh number (104≤Ra≤106), Hartmann number (0≤Ha≤60), hybrid nanoparticles volume fraction (0≤ϕhnp≤0.04), orientation of magnetic field (00≤γ≤600) and magnetic number (0≤δ0≤1) on the hybrid nanofluid flow and energy transfer are analyzed. Obtained results indicate that the combinations of the heaters location and the shaped of the geometry have major impact on all the transport phenomena. Moreover, Cu–Al2O3 hybrid nanoparticles play an essential role to levitate the thermal transmission compared to other physical parameters. An addition of hybrid nanoparticles leads to an increase in the average Nusselt number up to 10.8% in Case-I, 12% in Case-II, 11.66% in Case-III and 11.65% in Case-IV. The novelty of the present work is to examine the rheological properties of hybrid nanoliquid in a novel shaped cavity subjected to magnetic field dependent viscosity (MFDV) along with flush mounted variable heaters.

Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions

In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.