Finite Difference Method

This chapter introduces the Finite Difference Method (FDM) as a foundational approach in Computational Fluid Dynamics (CFD), emphasizing how conservation equations are discretized on structured grids using Taylor-series-based difference formulas. It begins with first-order forward and backward differences and progresses to second-order central and higher-order approximations. A central theme is the discussion on Compact Finite Difference Schemes (KFD), including Padé schemes and Lele&s;s compact formulations, which offer spectral-like resolution by implicitly coupling neighboring derivative values. Optimized High-Order Compact (OHOC) schemes, such as the Kim–Lee 8th-order scheme, are highlighted for their superior phase accuracy. Modified wavenumber analysis is used to assess how closely a numerical scheme captures wave behavior, especially regarding phase error (dispersion) and numerical diffusion (dissipation). Aliasing effects and the Nyquist wavenumber are discussed. Finally, the chapter addresses finite differences on non-uniform grids using coordinate transformations and corrections to retain accuracy across stretched meshes.