The acoustic wave equation is an important basis for seismic wave propagation, imaging, and migration. This equation has led to the development of finite difference methods, but these methods are prone to numerical instability and grid dispersion problems, with the issue of dispersion being the most vital. Optimization methods can improve the accuracy across a large range of wavenumbers or frequencies. In this paper, we present a novel evolutionary optimization scheme for the acoustic wave equation. Our approach involves combining the fruit fly optimization algorithm (FOA) and sampling approximation (SA) to obtain the optimal difference operator for a wide range of wavenumbers. The difference coefficients are optimized to be extracted by the FOA, and the function of fitness, which is used to introduce an iterative process to determine the best smells, is evaluated by acoustic wave simulations. Based on the space-domain dispersion relation, we prove that the accuracy with which the absolute error is minimized is the same as that of the relative error. Within a given range of wavenumbers, we propose a fitness function by minimizing the absolute errors of the space-domain dispersion relation, and the dispersion analysis reveals that this scheme is superior to the Taylor-expansion scheme. We conduct two experiments by applying homogeneous and complex models, respectively. Further, the modeling results indicate that the fruit fly optimization approach in conjunction with sampling approximation preserve a higher accuracy for a small operator length and reduce the numerical artifacts.

The staggered grid finite difference method has emerged as one of the most commonly used approaches in finite difference methodologies due to its high computational accuracy and stability. Inevitably, discretizing over time and space domains in finite difference methods leads to numerical artifacts. This paper introduces a novel approach that combines the widely used Taylor series expansion with the least squares method to effectively suppress numerical dispersion. We have derived the coefficients for the staggered grid finite difference method by integrating Taylor series expansions with the least squares method. To validate the effectiveness of our approach, we conducted analyses on accuracy, dispersion, and stability, alongside simple and complex numerical examples. The results indicate that our method not only inherits the capabilities of the original Taylor series and least squares methods in suppressing numerical dispersion across small and medium wavenumber ranges but also surpasses the original methods. Moreover, it demonstrates robust dispersion suppression capabilities at high wavenumber ranges.

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balance property. Instead of using a five-point stencil in the same fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two- dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (J. Sci. Comput., 85:29, 2020), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient and higher resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and greater efficiency than the classical fifth-order finite difference WENO scheme of Jiang and Shu. Various benchmark numerical examples are presented to show the fifth-order accuracy, great efficiency, high resolution and robustness of the proposed HWENO scheme.