In the dissertation, we mainly consider developing efficient numerical schemes for Allen-Cahn and Cahn-Hilliard equations, which are the origin of the phase-field equations.In the first part of the disseration, we present a new solver for nonlinear second-order parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution, which facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. We also solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions.In the second part of the disseration, we extend our Method of lines transpose (MOLT) scheme to Cahn-Hilliard (CH) and vector Cahn-Hilliard (VCH) equations. Our first step is to establish the gradient stability for CH models. This procedure is just a simple change of variables where one of the fixed points is subtracted from the original variable. We prove that in the semi-analytic setting, using Backwards Euler time stepping.After discretizing in time, we proceed to our spatial solver for inverting the linear part of the semi-analytic operator onto the non-linear part to construct an efficient fixed point method. This is done by factoring the fourth into the modified Helmholtz operators, which defined above. By including the splitting error into the right hand side of the fixed-point method, we arrive at a non-split scheme. We also combine the MOLT formulation with existing time stepping for high order time stepping methods, and numerically demonstrated the gradient stable property in 1D and 2D in all simulations run. Time adaptive methods are shown to be more efficient than using the same method with large fixed time steps.
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