In wave propagation problems, finite difference methods implemented on staggered grids are commonly used to avoid checkerboard patterns and to improve accuracy in the approximation of short-wavelength components of the solutions. In this study, we develop a mimetic finite difference (MFD) method on staggered grids for transport operators with divergence-free advective field that is proven to be energy-preserving in wave problems. This method mimics some characteristics of the summation-by-parts (SBP) operators framework, in particular it preserves the divergence theorem at the discrete level. Its design is intended to be versatile and applicable to wave problems characterized by a divergence-free velocity. As an application, we consider the electrostatic shear Alfv\'en waves (SAWs), appearing in the modeling of plasmas. These waves are solved in a magnetic field configuration recalling that of a tokamak device. The study of the generalized eigenvalue problem associated with the SAWs shows the energy conservation of the discretization scheme, demonstrating the stability of the numerical solution.

翻译：在波传播问题中，在交错网格上实现的有限差分方法通常用于避免棋盘格模式，并提高解中短波分量的逼近精度。本研究针对无散对流场的输运算子，发展了一种交错网格上的拟有限差分方法，该方法被证明在波问题中具有能量守恒性。该方法模拟了分部求和算子框架的某些特性，特别是在离散层面保持了散度定理。其设计旨在具有通用性，适用于以无散速度为特征的波问题。作为应用，我们考虑了等离子体建模中出现的静电剪切阿尔芬波。这些波在模拟托卡马克装置的磁场构型中求解。对与剪切阿尔芬波相关的广义特征值问题的研究表明，离散化格式具有能量守恒性，证明了数值解的稳定性。