Stability and convergence analysis for fully discrete
 pseudo spectral numerical schemes to nonlinear PDEs are
 presented in this talk, such as viscous Burgers' equation
 and incompressible Navier-Stokes equations.
 Related applications to incompressible Euler equation and
 quasi-geostrophic equation will also be addressed, in both
 2-D and 3-D, for smooth and vortex sheet initial data.
 In addition, high order time stepping schemes, including
 Adams Bashforth-Adams Moulton multi-step schemes up to
 fourth order accuracy and high order explicit SSP schemes,
 will be explored in detail. Unconditional stability
 is established for the implicit time stepping algorithms