Variable Stepsize, Variable Order Methods for Partial Differential Equations

Variable stepsize, variable order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in complex applications due to their computational complexity and the difficulty to implement them in legacy code. The goal of this dissertation is to develop, analyze, and test new VSVO methods that have the same computational complexity as their nonadaptive counterparts per step. Adaptivity allows these methods to take fewer steps, which makes them globally less complex. Herein, we show how to use any backward differentiation formula (BDF) method as the basis for a VSVO method. Order adaptivity is achieved using an inexpensive post-processing technique known as time filtering. Time filters do not add to the asymptotic complexity of these methods, and allow for every possible order in the VSVO family to be computed for the same cost as one BDF solve. This approach yields new, nonstandard timestepping methods that are not in the literature, and we analyze their stability and accuracy herein. Backward Euler (BDF1) and BDF2 are extremely ubiquitous methods, and this research demonstrates how they can be converted to order adaptive codes with only a few additional lines of code. We also develop a solver called Multiple Order One Solve Embedded 2,3,4 (MOOSE234). MOOSE234 is a VSVO method based on BDF3 that computes approximations of order two, three and four each step. All three approximations in MOOSE234 are at least A(alpha) stable, and the second order approximation is A stable. While these methods are generally applicable to any system that is first order in time, we focus on issues pertaining to the Navier-Stokes equations. Our methods have been optimized for Navier-Stokes solvers, and we include linearly implicit and implicit-explicit (IMEX) versions.