Efficient spectral methods and stable time discretizations for a class of parabolic type PDES

This thesis focuses on numerical methods for a class of nonlinear parabolic type PDEs in materials science, fluid dynamics, and finance. My contribution in the spatial discretization consists of two parts. First, an efficient spectral-Galerkin methods for systems of coupled second-order equations is proposed. Second, a GPU parallelized spectral collocation method for 2-D elliptic equations is designed and implemented. For the time discretization, I propose energy stable schemes for anisotropic Cahn-Hilliard systems in crystalline growth, a linear and energy stable scheme for the Navier-Stokes-Cahn-Hilliard system in the moving contact line problem, and second order stable schemes for the Merton jump diffusion model in European option pricing. I also prove the stability and convergence for the first-order rotational velocity correction projection method for the incompressible Navier-Stokes equation