This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, $\Delta p\pm {\lambda }^{2}p=f$. The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of $\lambda$-values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable $\lambda$-values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.

本文介绍了一种用于推导三维Helmholtz方程的高阶有限差分格式的新方法， $\Delta p\pm {\lambda }^{2}p=F$。新方法可以达到八阶精度。它基于隐式表示，同时计算未知变量及其对应的导数。该方案是从泰勒级数展开和平面波理论分析获得的，它是通过对标准中心有限差分进行的一些修改而构造的。给出了经典的色散分析以及数值波数与精确波数之间的误差。提出了数值实验，以验证所提方法在广泛范围内的可行性和准确性。$\lambda$值。而且，数值方法能够高精度地解决变数的单调和振荡亥姆霍兹问题。$\lambda$值。因此，所提出的公式产生了一种有吸引力的方法，易于实施，具有很高的精度，但计算成本与显式公式几乎相同。