# Brian Wetton

#### Relevant Thesis-Based Degree Programs

#### Affiliations to Research Centres, Institutes & Clusters

## Graduate Student Supervision

##### Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

In this thesis we study how to design accurate, efficient and structure-preserving numerical schemes for phase field models including the Allen–Cahn equation, the Cahn–Hilliard equation and the molecular beam equation. These numerical schemes include the explicit Runge–Kutta methods, exponential time differencing (ETD) Runge–Kutta methods and implicit-explicit (IMEX) Runge–Kutta methods. Note that the phase field models under consideration are gradient flows whose energy functionals decrease with time. For the Allen– Cahn equation, it is well known that the solution satisfies the maximum principle; for the Cahn–Hilliard equation, although its solution does not satisfy the maximum principle, the solution is also bounded in time. When designing numerical schemes, we wish to preserve certain stabilities satisfied by the physical solutions. We first make use of strong stability preserving (SSP) Runge-Kutta methods and apply some detailed analysis to derive a class of high-order (up to 4) explicit Runge-Kutta methods which not only decrease the discrete energy but also preserve the maximum principle for the Allen–Cahn equation. Secondly, we prove that the second-order exponential time differencing Runge-Kutta methods decrease the discrete energy for the phase field equations under investigation. Moreover, it can be shown that the ETDRK methods can also preserve the maximum bound property for the Allen–Cahn equation. What is more important is that both properties are preserved unconditionally, in the sense that the stability conditions do not depend on the size of time steps. Although the proof is only valid for second-order schemes and still open for higher-order methods, its numerical efficiency has been well observed in computations. The third approach is the implicit-explicit (IMEX) Runge–Kutta (RK) schemes, i.e. taking the linear part in the equation implicitly and the nonlinear part explicitly when solving. A class of high-order IMEX-RK schemes are studied carefully. We demonstrate that some of the IMEX-RK schemes can preserve the energy decreasing property unconditionally for all the phase-field models under investigation.

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In this dissertation, we study analytical and numerical methods on three topics in the area of partial differential equations (PDE). These topics are: the Allen-Cahn dynamics (AC) in the study of phase field models for materials science problems, the Oxygen depletion model (OD) in the study of free boundary problems, and the stationary surface quasi-geostrophic equation (SQG) in the study of fluid dynamics. We first study the behaviour in the meta-stable regime of AC and show by computation evidence and asymptotic analysis that backward Euler method satisfies energy stability with large time steps. We also give a rigorous proof for the two-dimensional radially symmetric case. In the second project, we show several mathematical formulations of OD from the literature and give a new formulation based on a gradient flow with constraint. We prove the equivalence of all formulations and study the numerical approximations of the problem that arise from the different formulations. More general (vector, higher order) implicit free boundary value problems are discussed. In the final project, we develop a new framework of ``convex integration scheme'' and construct a non-trivial solution to the stationary SQG. We thus prove the non-uniqueness of the solutions to the stationary SQG.

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In this thesis we present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in ℝ². We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities. In parallel with the generalized numerical method, we present some analytical solutions to the GMS model on a circular and slightly perturbed circular curve geometry. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle we show that there are two distinct perturbations, one resulting from the introduction of a angular dependent radius, and one caused by Fourier mode interactions which causes a vertical shift to the solution. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.

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Magnetized target fusion reactors are a modern idea to generate hydrogen fusion energy on earth. The design entails confining a plasma with a magnetic field and crushing it in an imploding shell of molten metal. Such a design has many unresolved questions in terms of its feasibility as a power source and ways to make it efficient. In this thesis, we will look into two of the approaches undertaken to explore these questions. Firstly, we use a coordinate transformation and implement a novel flux-limited, split-step, finite volume scheme for nonlinear coupled hyperbolic partial differential equations. With this numerical scheme, we do a parameter sensitivity analysis for the design performance. Secondly, by a careful series of asymptotic arguments, we establish a leading order asymptotic expression for the plasma compression. This expression is qualitatively consistent with the numerical work, but it also gives new insights into how the device operates. Together these approaches allow us to infer key design parameters for the success of magnetized target fusion. We will conclude with a look into the viability of magnetized target fusion and some problems for future work.

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No abstract available.

##### Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

The pseudo-two-dimensional (P2D) model is a complex mathematical model that can capture the electrochemical process in Li-ion batteries. However, simulation of the model also brings a heavy computational burden. Many simplifications to the model have been introduced in the literature to reduce the complexity. We present methods for fast computation of two models: an asymptotically scaled P2D model and the full P2D model. The techniques developed in this work can be used when simplifications are not accurate enough. By rearranging the calculations, we reduce the complexity of the linear algebra problem. We also employ automatic differentiation, using the open source package JAX, for robustness, which also allows easy implementation of changes to coefficient expressions. The method alleviates the computational bottleneck in P2D models without compromising accuracy.

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Lithium-ion batteries are ubiquitous in modern society. The high powerand energy density of lithium-ion batteries compared to other forms of electrochemical energy storage make them very popular in a wide range of applications, most notably electric vehicles (EVs) and portable devices suchas mobile phones and laptop computers. However, despite the numerousadvantages of lithium-ion batteries over other forms of energy sources, theirperformance and durability still suffer from aging and degradation. The purpose of the work presented in this thesis is to investigate how different loadcycle properties affect the cycle life and aging processes of lithium-ion cells.To do so, two approaches are taken: physical experiments and mathematicalmodeling.In the first approach, the cycle life of commercial lithium-ion cells ofLiNiCoAlO₂ chemistry was tested using three different current rates to simulate low-, medium-, and high-power consuming applications. The batteries are discharged/charged repeatedly under the three conditions, all whiletemperature, voltage, current, and capacity are recorded. Data arising fromthe experiments are then analyzed, with the goal of quantifying batterydegradation based on capacity fade and voltage drop. The results are thenused to build two predictive models to estimate lithium-ion battery state-of-health (SoH): the decreasing battery V₀₊ model and the increasing CVcharge capacity model. Furthermore, a simple thermal model fitted fromthe battery temperature profiles is able to predict peak temperature under different working conditions, which may be the solution to temperaturesensitive applications such as cellphones.The limitation to physical experiments is that they can be costly andextremely time-consuming. On the other hand, mathematical modeling andsimulation can provide insight, such as the internal states of the battery,that is either impractical or impossible to find using physical experiments.Examples include lithium-ion intercalation and diffusion in electrodes andelectrolytes, various side-reactions, double-layer effects, and lithium concentration variations across the electrode layer. Thus, in the second approach,work focuses on implementing the pseudo-two-dimensional (P2D) model, themost widely accepted electrochemical model on lithium-ion batteries. TheP2D model comprises highly-nonlinear, tightly-coupled partial differentialequations that calculate lithium concentration, ionic flux, battery temperature and potential to significant accuracy. The unparalleled prediction abilities of the P2D model, however, are shadowed by the low computationalefficiency. Thus, much of this work focuses on reducing model complexity toshorten effective simulation time, allowing for use in applications, such as abattery management system, that have limited computational resources. Inthe end, four model reductions have been identified and successfully implemented, with each one achieving a certain standard of accuracy.

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It is well known that Allen-Cahn equation and Cahn-Hilliard equation are essential to study the phase separation phenomenon of a two-phase or a multiple-phase mixture. An important property of the solutions to those two equations is that the energy functional, which is defined in this thesis, decreases in time. To study these solutions, researchers developed different numerical schemes to give accurate approximations, since analytic solutions are only available in a very few simple cases. However, not all schemes satisfy the energy-decay property, which is an important standard to determine whether the scheme is stable. In recent work, Li, Qiao and Tang developed a semi-implicit scheme for the Cahn-Hilliard equation and proved the energy-decay property. In this thesis, we extend the semi-implicit scheme to the Allen-Cahn equation and fractional Cahn-Hilliard equation with a proof of the energy-decay property. Moreover, this semi-implicit scheme is practical and could be applied to more general diffusion equations while preserving the energy-decay stability.

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In some nonlinear reaction-diffusion equations of interest in applications, there are transition layers in solutions that separate two or more materials or phases in a medium when the reaction term is very large. Two well known equations that are of this type: The Allen-Cahn equation and the Cahn-Hillard equation. The transition layers between phases evolve over time and can move very slowly. The models have an order parameter epsilon. Fully developed transition layers have a width that scales linearly with epsilon. As epsilon goes to 0, the time scale of evolution can also change and the problem becomes numerically challenging. We consider several numerical methods to obtain solutions to these equations, in order to build a robust, efficient and accurate numerical strategy. Explicit time stepping methods have severe time step constraints, so we direct our attention to implicit schemes. Second and third order time-adaptive methods are presented using spectral discretization in space. The implicit problem is solved using the conjugate gradient method with a novel preconditioner. The behaviour of the preconditioner is investigated, and the dependence on epsilon and time step size is identified. The Allen-Cahn and Cahn-Hilliard equations have been used extensively to model phenomena in materials science. We strongly believe that our high order adaptive approach is also easily extensible to higher order models with application to pore formation in functionalized polymers and to cancerous tumor growth simulation. This is the subject of ongoing research.

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The removal of artificially and naturally placed oils found in the ground is of paramount concern for environmental reasons and for the extraction of crude oil for energy. While many companies are involved in this soil remediation process, the physics of the underlying problem are not very well understood. We present a mathematical model and analysis of oil extraction in a porous soil based around the injection of water. We first consider a saturation analysis based on conservation of mass between oil and water in a one-dimensional setting and we find that based on certain parameter values, the water-oil interface goes unstable producing viscous finger patterns. We then include the effects of surface tension between oil and water to determine how this affects the growth of such fingers. We conclude that in the limit of small surface tension effects, the results generalize to the original problem, but more importantly, we deduce that there is a scaling which places the formulation into a setting that is invariant with respect to the magnitude of surface tension effects. With this scaling, we notice that the effect of surface tension is to limit the growth of fingers to a maximal wave number and to prevent their formation entirely beyond a certain critical wave number. Finally with the inclusion of temperature, via heated water injection, we see the formation of a dual fingering pattern: one associated with the mass conservation analysis of the oil-water interface and one associated with the conservation of energy across an interface where thermal gradients occur. We see that the thermal gradients across the interface where temperature drops induce unstable viscous patterns with a higher wave number than would occur for an equivalent isothermal interface where there was solely a change in viscosity. The thermal gradients also promote fingering development downstream across the classical viscosity differential driven interface but at the expense of lowering the interfacial velocity. It is interesting to note that the change in saturation that occurs across the energy interface is a result of a pseudo-free boundary created by the thermal problem.

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This thesis is a combination of two research projects in applied mathematics, which use the applied math techniques of numerical and asymptotic analysis to study real-world problems. The first problem is in superconductivity. This section is motivated by recent experimental results at the Paul Sherrer Institute. Here, we need to determine how the surface roughness of a superconductor influences the penetration properties of an externally applied magnetic field. We apply asymptotic analysis to study the influences, and then verify the accuracy - even going well-beyond the limits of the asymptotics - by means of computational approximations. Through our analysis, we are able to offer insights into the experimental results, and we discover the influence of a few particular surface geometries. The second problem is in gas diffusion. The application for this study is in fuel cells. We compare two gas diffusion models in a particular fuel cell component, the gas diffusion layer, which allows transport of reactant gases from channels to reaction sites. These two models have very different formulations and we explore the question of how they differ qualitatively in computing concentration changes of gas species. We make use of asymptotic analysis, but also use computational methods to verify the asymptotics and to study the models more deeply. Our work leads us to a deeper understanding of the two models, both in how they differ and what similarities they share.

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