Compact finite difference schemes and error estimation for third-order Emden-Fowler equations

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 reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

Sixth order compact finite difference schemes for Poisson interface problems with singular sources

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations

Recently, the finite difference method, a tool for wave acoustic simulations, can be applied to practical analysis. However, the numerical dispersion which makes propagation speed in simulations change according to the wave length, is a severe problem to maintain high accuracy in the analysis. The numerical error accompanied by the time integration hinders to carry out an accurate simulation. An examination of a compact finite difference scheme which can approximate differentiation precisely on a comparatively large grid spacing was made. It was shown that the adjusting the coefficients in the compact finite difference scheme can simulate a wave propagation accurately with a small amount of computation. The improvement method of the time integration was also discussed. The possibility of improvement of accuracy through a multi‐step integration is shown.

In this article, two sets of fourth‐order compact finite difference schemes are constructed for solving heat‐conducting problems of two or three dimensions, respectively. Both problems are with Neumann boundary conditions. These works are extensions of our earlier work (Zhao et al., Fourth order compact schemes of a heat conduction problem with Neumann boundary conditions, Numerical Methods Partial Differential Equations, to appear) for the one‐dimensional case. The local one‐dimensional method is employed to construct these two sets of schemes, which are proved to be globally solvable, unconditionally stable, and convergent. Numerical examples are also provided. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

In this article, for a class of linear systems arising from the compact finite difference schemes, we apply Krylov subspace methods in combination the ADI, BLAGE,... preconditioners. We consider our scheme in solution of hyperbolic equations subject to appropriate initial and Dirichlet boundary conditions, where is constant. We show, the BLAGE preconditioner is extremely effective in achieving optimal convergence rates. Numerical results performed on model problems to confirm the efficiency of our approach.

This paper investigates implementation of upwind compact implicit and explicit high-order finite difference schemes for solution of the level set equation. The upwind compact implicit and explicit high-order finite difference schemes are well-known techniques to descritize spatial derivatives for convection term in hyperbolic equations. Applying of upwind high-order schemes on the level set equation leads to less error and CPU time reduction compared to essential non-oscillatory (ENO), weighted essential non-oscillatory schemes (WENO), and even different particle level set methods. The results indicate the error based on area loss decreases drastically with applying high-order upwind, especially implicit schemes compared to ENO and WENO schemes. In addition, CPU time is reduced compared to these methods. Hence, the upwind high-order schemes are suggested as excellent alternatives for the previous schemes to calculate spatial derivatives in the level set equation.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

This study investigates the propagation of nonlinear heat waves in a rigid thermal conductor, where classical models do not accurately capture high-frequency thermal behavior. To address this, we propose a compact finite difference method (CFDM) with spectral–like resolution. The model incorporates non-linear thermal conductivity and a temperature-dependent second sound speed. Using a Padé–based compact scheme, we discretize the governing equations and analyze stability through the von Neumann method, confirming unconditional stability. Numerical simulations demonstrate the high accuracy, rapid convergence, and reduced numerical oscillations of the method compared to existing approaches. Furthermore, an error estimation framework based on Sobolev space norms supports the theoretical convergence claims. These results establish CFDM as an efficient and robust tool for modeling non-linear thermal wave phenomena.

Higher-order compact finite difference scheme with multigrid algorithm is applied in this paper for solving one-dimensional and two-dimensional inhomogeneous Helmholtz equations. In two-dimensional case, the suggested scheme has the stencil of twenty one points. An efficient solver multigrid method yields eighth-order accurate approximation on both fine and coarse grids. For the Neumann boundary condition, an eighth-order accurate representation is also developed. The accuracy and efficiency of eighth-order compact difference scheme are exhibited through graphical illustrations and computed results are drafted in tabular form to validate the numerical experiments.

In this paper, we investigated an advantage of the finite difference lattice Boltzmann method (FDLBM) to direct numerical simulations of sound propagation including scattering by a body. In order to incorporate, monopole sound source into the flow, a new FDLB formulation was proposed using the Chapman-Enskog expansion. We calculated the acoustic scattering problem around a circular cylinder using the FDLBM discretized by non-compact finite difference scheme which is faster than compact finite difference scheme. The results agreed with both the exact solution and the accurate calculation by the macroscopic equations based on the discretizations of compact finite difference scheme. As a result, the FDLBM was proved to have a merit to a calculation of sound propagation. Also, we estimated error of the FDLBM in the acoustic pulse problem.

In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schrodinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time.

Efficient semi-implicit compact finite difference scheme for nonlinear Schrödinger equations on unbounded domain