Optimization of compact finite difference method for wave acoustic simulation

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

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.

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

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.

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.

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.

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

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

We have developed compact implicit finite-difference (FD) schemes in the time-space domain based on the second-order FD approximation for accurate solution of the acoustic wave equation in 1D, 2D, and 3D. Our method is based on the weighted linear combination of the second-order FD operators with different spatial orientations to mitigate numerical error anisotropy and the weighted averaging of the mass acceleration term over the grid points of the second-order FD stencil to reduce the overall numerical dispersion error. We have developed a derivation of the schemes for 1D, 2D, and 3D cases. We obtain their corresponding dispersion equations, then we find the optimum weights by optimization of the time-space domain dispersion function, and finally we tabulate the optimized weights for each case. We analyze the numerical dispersion, stability, and convergence rates of our schemes and compare their numerical dispersion characteristics with the standard high-order ones. We also discuss the efficient solution of the system of equations associated with our implicit schemes using the conjugate-gradient method. The comparison of dispersion curves and the numerical solutions with the analytical and the pseudospectral solutions reveals that our schemes have better performance than the standard spatial high-order schemes and remain stable for relatively large time steps.

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.

On consistent boundary closures for compact finite-difference WENO schemes

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 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.

The high-order compact finite difference (HCFD) method is adapted for interconnect modeling. Based on the compact finite difference method, the HCFD method employs the Chebyshev polynomials to construct the approximation framework for interconnect discretization, and leads to improved equivalent-circuit models. The HCFD-based modeling requires far fewer intervening grid points for building an accurate discrete model of the transmission line than other numerical methods like traditional Finite Difference (FD) method. It is believed that given the number of state variables, the presented method gives more accurate results than other known passive discrete modeling methods. The theoretical proof shows that HCFD-based modeling preserves the passivity.