Numerical Simulation of the Generalized Wave Equation Using an Improved Adaptive Differential Evolution–Optimized Staggered-Grid Finite-Difference Scheme

The generalized wave equation expands traditional elastic wave theory by incorporating couple-stress and strain-gradient effects, providing a more comprehensive description of subsurface mechanical behavior. This formulation is particularly suited for capturing rotational motion and scale-dependent phenomena induced by microstructural heterogeneity. In this study, the generalized wave equation with couple-stress and strain-gradient terms is derived and numerically solved using a finite-difference (FD) approach, selected for its simplicity, efficiency, and parallelizability. However, traditional FD schemes often exhibit numerical dispersion. To address this, we develop an optimized staggered-grid finite-difference operator guided by an enhanced self-adaptive differential evolution (SADE) algorithm. The algorithm employs a multi-strategy adaptive mutation mechanism and a dynamically updated strategy pool to balance global exploration with local exploitation. The resulting first-order spatial derivative operators achieve high accuracy, markedly reducing numerical dispersion. Numerical tests on layered and Marmousi models confirm that the proposed method more faithfully reproduces wavefield phase and amplitude characteristics than traditional FD method and other metaheuristic optimization techniques. This work not only establishes a novel numerical optimization framework but also provides an effective computational tool for high-fidelity, seismic wave simulation, imaging, and inversion.