High-accuracy difference and meshless methods for fluid and solid mechanics

Two types of high-accuracy methods for PDE’s are outlined. The first one is the Compact Differencing (CD) methods of a fixed high-order and their extension to multioperators-based both noncentered and centered schemes. The techniques show their peak performance in the cases of relatively simple geometries. As an illustration, a shear layer instability problem with vorticity rolling-up is considered. For complicated geometries, large deformations etc., a meshless method based on using Radial Basis Functions in a “finite difference mode” has been developed. Numerical examples are presented, the main emphasize being placed on solid mechanics problems.