Development of a fast and accurate time stepping scheme for the functionalized Cahn-Hilliard equation and application to a graphics processing unit

This dissertation explores and develops time-stepping schemes for computing solutions to the Functionalized Cahn-Hilliard (FCH) model. It is important to find a scheme that is both fast enough to compute evolution to the long-time states and to give enough accuracy to capture important geometric events. The FCH model is relatively new, and very little work has been done to develop efficient numerical schemes for its simulation, so much of this work is based on the extensive work done on the Cahn-Hilliard (CH) model. For each of the methods, the spatial approximation is computed with a Fourier spectral method. All of the schemes are adapted to be computed on a graphics processing unit (GPU) which gives significant improvements in the speed of the simulation.;First, an implicit-explicit (IMEX) method will be introduced that is based on a convex splitting of the right hand side of the equation. This splitting guarantees that the solutions will decay in energy for any size time step, which gives numerical stability for very large time steps. With this splitting, a novel iterative method for solving the implicit portion greatly improves the numerical efficiency.;Second is the development of a fully implicit method that attains high accuracy. The method uses a conjugate gradient method to solve the Netwon's method iterations, and is preconditioned using a physics based approximation to the operator that is easy to invert and numerically efficient.;Lastly, exponential time differencing (ETD) methods are derived for the Cahn-Hilliard and Functionalized Cahn-Hilliard Equations. The ETD methods are all explicit which affords computation speed, and higher order versions are natural extensions giving accurate time stepping.;Finally, numerical experiments for the three types of methods compare the accuracy and speed. These simulations are performed for both fixed time-step simulations as well as adaptive time steps. This gives a clear picture of the strengths and weaknesses, and it gives enough information to determine which time-stepping method will work best for approximating solutions to the FCH equation.