| This thesis contains two main topics related to high-order numerical approximations.; The first part is titled High-Order Implicit-Explicit Runge-Kutta Time Integration Schemes. Despite the popularity of high-order Explicit Runge-Kutta (ERK) methods for integrating semi-discrete systems of equations, such as the classical fourth-order scheme, they suffer from severe stability-based time step restrictions for problems with high levels of geometry-induced stiffness or operator/physics-induced stiffness. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems.; The second part is titled Time-Consistent Filtering in Spectral Methods. The comparison of numerical results for implicit-explicit and fully explicit Runge-Kutta time integration methods for a nozzle flow problem shows that filtering can sig nificantly degrade the accuracy of the numerical solution for longtime integration problems. We demonstrate analytically and numerically that filtering-in-time errors become additive for || uN(x, t + kDelta t) - uN(x, t)|| << ||uN( x, t)|| when nonconsistent filters are used, and suggest the development and implementation of time-consistent filters. |
