Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive. The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze. The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing. / Doctor of Philosophy / Almost all time-dependent physical phenomena can be effectively described via ordinary differential equations. This includes chemical reactions, the motion of a pendulum, the propagation of an electric signal through a circuit, and fluid dynamics. In general, it is not possible to find closed-form solutions to differential equations. Instead, time integration methods can be employed to numerically approximate the solution through an iterative procedure. Time integration methods are of great practical interest to scientific and engineering applications because computational modeling is often much cheaper and more flexible than constructing physical models for testing. Large-scale, complex systems frequently combine several coupled processes with vastly different characteristics. Consider a car where the tires spin at several hundred revolutions per minute, while the suspension has oscillatory dynamics that is orders of magnitude slower. The brake pads undergo periods of slow cooling, then sudden, rapid heating. When using a time integration scheme for such a simulation, the fastest dynamics require an expensive and small timestep that is applied globally across all aspects of the simulation. In turn, an unnecessarily large amount of work is done to resolve the slow dynamics. The goal of this dissertation is to explore new "multimethods" for solving differential equations where a single time integration method using a single, global timestep is inadequate. Multimethods combine together existing time integration schemes in a way that is better tailored to the properties of the problem while maintaining desirable accuracy and stability properties. This work seeks to overcome limitations on current multimethods, further the understanding of their stability, present new applications, and most importantly, develop methods with improved efficiency.

Three- and four-derivative k-step Hermite-Birkhoff-Obrechkoff (HBO) methods are constructed for solving stiff systems of first-order differential equations of the form y'= f(t,y), y(t0) = y0. These methods use higher derivatives of the solution y as in Obrechkoff methods. We compute their regions of absolute stability and show the three- and four-derivative HBO are A( 𝜶)-stable with 𝜶 > 71 ° and 𝜶 > 78 ° respectively. We conduct numerical tests and show that our new methods are more efficient than several existing well-known methods.