A high-resolution finite difference scheme with neural network optimized dispersion and localized adaptive dissipation

High-order accuracy and high-resolution numerical schemes have been extensively utilized in computational fluid dynamics (CFD) and computational aeroacoustics (CAA) for their ability to precisely capture wave shapes. This paper introduces a novel finite difference scheme, the Neural Network Optimized Dispersion and Localized Adaptive Dissipation (NNoD-LAD) scheme, which combines a dispersion model and a dissipation model to achieve fourth-order accuracy and exceptional resolution. The NNoD-LAD's dispersion model is designed to address different wavenumber ranges effectively. For the low-intermediate wavenumber range, the dispersion coefficient is set to ensure zero dispersion error, guaranteeing accuracy. In the high wavenumber range, a fully connected neural network optimizes the coefficient to minimize dispersion error while maintaining numerical stability. The dissipation model in the NNoD-LAD scheme adjusts dissipation flexibly based on the wavenumber and solution variation. This improves simulation capabilities in the low-intermediate wavenumber range and effectively suppresses spurious oscillations in the high wavenumber range. Additionally, a new scale information recognizer has been developed to accurately compute the local wavenumber, at half-point locations. Numerical tests demonstrate that the NNoD-LAD scheme offers higher resolution compared to previous spectrally optimized schemes, making it a promising tool for CFD and CAA applications.