### Numerical Methods

Numerical methods for solving differential equations, such as equations that arise in physics, fall into the broad categories of time integration and spatial methods. In particular, **Integral Deferred Correction (IDC) methods** are high order numerical time integrators whose structure leads to simple construction of arbitrary order integrators. IDC methods involve a low order prediction step and correction of the prediction to higher order. A key component is the solution of an error equation in integral form which stabilizes the method.

Certain modifications to IDC allow high order numerical solutions to multi-scale and/or nonlinear problems in plasma physics, such as the Vlasov-Poisson (VP) system. These modifications include incorporating **semi-implicit methods** to solve an initial value problem (arising from method of lines discretization) with a stiff and nonstiff term, or employing **operator splitting** into IDC’s prediction and correction steps. Incorporation of split IDC with conservative semi-Lagrangian WENO interpolation results in well-resolved solutions to classic plasma physics problems such as Landau damping and two stream instability. Further, IDC methods can be implemented with adaptive time stepping and in parallel. With continued progress in fusion energy research, such numerical methods continue to be valuable.

### Reliability Analysis

Reliability analysis is used in engineering design to incorporate the real-life uncertainty and variability in materials, components, and systems through probabilistic rather than deterministic methods. It involves using statistical and probabilistic methods to predict the probability of failure and probability of safety, or reliability, of a structure or product.

For example, a steel bar may have variability in its material properties and strength due to uncontrollable manufacturing factors. The load that the steel bar will have to bear in its future use could be known to vary in time and circumstance as well as be imprecisely known. One could represent the material properties, load, and strength as random variables. A performance function of those variables is negative when the system fails. In the very simplistic linear case where the performance function equals strength minus load, failure occurs when the load is greater than the strength of the bar. Strength and load each have their own distribution. While the expected values of strength and load may indicate product reliability, an overlap of their distributions may occur, resulting in a probability of failure. The probability of failure may be found through analytic or, more often, numerical integration. In practice, the performance function may be nonlinear and/or unknown, and the distributions of the random variables may also be unknown. Experimental data may be sparse, so simulations such as Monte Carlo methods are utilized. I applied these standard techniques to develop research software in Python.

### Fuel Cell Technology

A fuel cell directly converts chemical energy from a fuel to electrical energy. Essentially, an electrical current is produced while hydrogen and oxygen ions meet through an electrolyte and combine to form water. The conversion is more efficient and cleaner than combustion.

Although this process is quite simple, the variety of materials and shapes of fuel cells, as well as the total system of preparing the fuel, managing the waste, and harnessing and distributing the resulting power, can be mathematically complicated. For example, multiple scales in both time and space occur in fuel cells, which result in stiff systems that present computational challenges.

Many existing mathematical investigations of fuel cells focus on the structure of the fuel cell itself, effectiveness of certain materials, efficiency of power production, and water management, but there is still much more to learn. For example, the study of fuel cell degradation over long periods of use is of interest.

Additional areas of vital research include infrastructure that supports fuel cell technology, such as hydrogen fuel production, storage, and distribution.