PSE and NPSE lie between the extremes of LST and DNS. The PSE and NPSE include nonparallel and curvature effects. For linear PSE (PSE), a single monochromatic wave is considered as the disturbance, which is decomposed into a rapidly varying “wave function” and a slowly varying “shape function”. If nonlinear effects are important, nonlinear PSE (NPSE) are derived in a fashion similar to PSE except that each disturbance quantity is transformed spectrally in the spanwise and temporal directions. Here each mode is the product of a “shape function” and a “wave function”. NPSE accommodates nonlinear interactions of multiple disturbances with small additional resource requirements compared to DNS. Used under appropriate physical and disturbance input conditions, the agreement between NPSE and experiments is remarkable.