Exponential propagation iterative methods of Runge-Kutta type (EPIRK) provide a flexible framework to derive efficient

exponential integrators for different types of ODE systems. Here we develop new classes of EPIRK methods depending on

the properties of the equations to be solved. We derive the order conditions for the new classes of integrators and present new approaches to solving the order conditions where we optimize coefficients to construct more efficient schemes. Performance comparisons are conducted on a set of test problems to demonstrate the computational advantages of the new schemes themselves and the overall improvement in computational cost due to the recent additions to the EPIRK family.

Exponential integrators have received renewed interest in recent years as a means to approximate stiff systems of ODEs, but are not currently widely used in high performance computing. There have been only limited performance studies comparing them to currently used methods, little work investigating how to optimize their design for computational efficiency, and almost no work on implementing and studying their performance on parallel computers. We present here a detailed performance breakdown and comparison of Krylov-based exponential integrators to each other and to Newton-Krylov implicit solvers, the currently most widely used class of methods for large-scale stiff problems. Our results show exponential integrators perform favorably compared to implicit integrators across a

number of different problems. We then introduce a new class of exponential integrators called exponential propagation iterative methods of Runge-Kutta type (EPIRK). Based on our performance analysis we consider some strategies for utilizing their structural features to construct schemes with improved computational eciency and demonstrate their effectiveness with some numerical experiments. We also describe a parallel implementation of a suite of exponential integrators and give some performance results which show encouraging performance of the methods on problems scaled up to thousands of processors when compared to CVODE, a production-grade parallel implementation of a Newton-Krylov implicit

integrators popularly used for high performance computing applications today. We conclude with consideration of possible future research directions.

Stiff systems of ordinary differential equations (ODEs) play an essential role in the temporal integration of many scientific and engineering problems. With the increase in complexity and size of these problems, it is crucial to have efficient time integration methods. In this dissertation, we focus on the construction, analysis, and applications of exponential time integrators. We address the efficiency of these methods in three main directions. First, we construct new efficient exponential multi-step methods. These methods are carefully derived to combine the accuracy of high-order schemes with the efficiency of low-order methods. We validate and apply the new schemes to a simplified atmospheric model and show that they can capture more details in the solution with a limited computational cost. Secondly, we develop a new ansatz for deriving partitioned implicit-exponential integrators. These methods are applied to problems where the forcing term of the system is comprised of stiff additive terms. We propose a new way of analyzing the stability properties of partitioned methods. Our results show that the increased accuracy and stability of our schemes offer improved efficiency compared to state-of-the-art methods. Finally, we introduce a new theoretical framework for deriving stiffly resilient exponential methods. This framework addresses the complexity of deriving exponential schemes and enforces advantageous properties. We establish an analytical solution for the order conditions used to derive methods. We derive a new class of exponential integrators and new schemes with an order of convergence higher than currently available methods. We also evaluate the performance of exponential time integration for the reduced magnetohydrodynamics equations. Our initial results focus on a simplified model capturing the essence of the full equations. We show that exponential integrators are a valuable alternative to the schemes currently used for this problem.

Computational combustion plays a major role in engineering applications. However, solving the large systems of differential equations that model combustive processes is challenging. One of the major computational difficulties with the numerical modeling of combustion lies in the widely varying time scales present in the model equations that describe chemical interactions of species with the fluid and thermodynamic transport phenomena. The resulting dramatic stiffness ofthe equations demands the development of more efficient temporal integrators to enable efficient and accurate combustion simulation.

Traditionally, the stiffness of computational combustion models has been addressed using implicit methods. However, the performance of the implicit schemes depends highly on the availability of an efficient preconditioner to alleviate stiffness constraints. Additionally, due to the complexity of the coupling between chemistry, fluid, and transport phenomena, splitting is often used to simplify the time integration of the model equations. Splitting, in turn, reduces the accuracy of theapproximation. While constructing a preconditioner for a portion of the source terms, such as chemical reactions, is feasible, this task becomes more complicated if one considers the full source term of the equations.

Exponential integrators have recently emerged as an efficient alternative to implicit methods for solving large-scale stiff systems, particularly when no effective preconditioner is available. In this thesis, we explore whether exponential methods can be used for combustion simulations and study the computational advantages of such schemes.

We first explore how to tackle the stiff nature of chemical kinetics in a model that forgoes transport phenomena to isolate the chemical kinetics terms and forms the core of various more complex combustion models. A novel time adaptive exponential integrator is presented and then applied to this zero-dimensional (Zero-D) combustion problem. We demonstrate that the new method can perform comparably to well-established implicit-Krylov time integration methods. We study the performance of the exponential integration methods and demonstrate how theyare affected by the spectrum of the problem.

We then extend the chemical kinetics core to include transport phenomena and develop an exponential integration-based numerical approach to the propagating flame front model in one dimension. In addition to the embedded homogeneousreactor problem, advection and diffusion terms are added, and continuity is considered, which generates a highly coupled system of PDEs. In these problems, operator splitting is typically used to separate the stiff chemical reaction source terms from the much slower transport phenomena. Effectively, this reduces the equations to a Zero-D problem and a transport problem, which needs to be solved at each time step. While effective, this introduces unfavorable splitting errors; we demonstrate that this split can be avoided when using an exponential time integration scheme. We study the performance of the new exponential integration approach for the model using several different chemical mechanisms and compare the performance of the new integrators to the state-of-the-art NGA code, which uses implicit methods.

Our computational models and numerical experiments indicate that exponential methods offer a promising approach to modeling combustion and highlight directions for further studies that can help to develop more efficient time integra-tors for computational combustion problems.

The problem of solving for charged particle dynamics under the influence of electromagneticfields is a fundamental component of particle simulation models to simulate plasma physics. This task - known as the particle pushing problem - is especially challenging when the plasma is strongly magnetized due to numerical stiffness arising from the wide range of time scales between highly oscillatory gyromotion and long term macroscopic behavior of the system, thus calling for computationally efficient numerical methods.

This dissertation investigates an alternative approach to numerically simulate strongly magnetized charged particle dynamics using exponential integration techniques. We first derive exponential integrators to solve strongly magnetized particle pushing problems and implement a novel algorithm to evaluate matrix φ functions that is computationally efficient for low dimensional problems. We then extend this work by deriving Nytröm type exponential integrators that effectively solve the particle pushing problem in its second-order form directly. Numerical experiments comparing these exponential integrators against two conventional particle pushing algorithms demonstrate that exponential integrators are superior for linear and weakly nonlinear problems and are competitive for nonlinear problems. In particular, the Nyström type exponential integrators exhibit significant computational savings over the standard exponential integrators. These results demonstrate that exponential integrators are a promising alternative to numerically solve strongly magnetized particle pushing problems.

We present our study of the behavior of water at the molecular level in the presence of an external electric field. We develop two model systems using the theory of molecular dynamics (MD) and demonstrate how the tendency of interfacial water molecules to reorganize explains the structural behavior of the systems.

Water molecules can interact with external electric fields. This interaction can be observed in various processes such as formation of raindrops in thunderclouds, lightning strikes and electrofreezing. Understanding of this interaction is relevant to variety of applications, like for example, electrospinning, electrospray ionization, crop spraying, spray painting, and inkjet printing. Furthermore, the interaction of water molecules with an external electric field plays an important role in the electroporation process.

First, we present a novel theory of electroporation that provides insight into initiation pore formation in biological cell membranes when they are exposed to an external electric field. Using thorough structural and energetic analysis, we demonstrate that the formation of a pore is driven by the reorganization of the interfacial water molecules into energetically favorable structures.

Second, we present our study of nano-sized water droplets placed in an external electric field. Our analysis exposes the molecular mechanism behind the nanodroplet shape change from a spheroid to a highly prolate ellipsoid. We demonstrate that a droplet extends its shape as water dipoles align with the electric field while simultaneously restructuring to minimize the dipole-related interaction energy. A new semi-empirical model is developed to predict the critical electric field value which separates the two states. We show that the new model sheds more light onto the dynamics of the system compared to previous theoretical results.

Finally, we conclude with a possible future research work.