The numerical error produced when computing slowly moving shocks in a finite volume framework is studied. The error is a result of discretizing a thin shock profile dynamic on the scale of spatial and temporal discretization. It has been studied in great detail by many practitioners, but no fix has been built to totally eradicate the problem unique to nonlinear hyperbolic systems. I argue that current shock capturing schemes do not accurately resolve the nonlinear shock profile and this results in erroneous oscillations. I study numerical tools used to pursue high-order accurate solutions of slow shocks, including a hybridized upwind-biased slope limiter for the piecewise parabolic method, a universal Osher Riemann solver, and a new weighted essentially non-oscillatory (WENO)-type slope limiter. The effectiveness of these tools are tested in two hyperbolic systems: the Euler equations and the ideal magnetohydrodynamics equations. The success of the upwind-biased slope limiter has been verified against high-order accurate schemes such as WENO.

A new interpolation method using an analytically solenoidal Gaussian Pro-

cess (GP) kernel has been introduced within the Unsplit Staggered Mesh (USM)

magnetohydrodynamics (MHD) solver, implemented in the University of Chicago

ASC FLASH Center’s FLASH4 code. USM computes numerically divergence-

free (up to machine accuracy) magnetic fields at the cell interface-centers at each

time step, and then uses those values to predict the cell-centered magnetic fields

through simple linear interpolation. The GP interpolation method presented here

(which replaces the simple linear interpolation) uses the computed cell interface-

centered magnetic fields as sample points to train a GP model with an analytically

divergence-free kernel function, which is used to predict these fields at the cell cen-

ters. The performance of the USM scheme using GP interpolation is studied and

compared to the original scheme using arithmetic averaging. Some improvements

can be seen, particularly when calculating the numerical divergence using the

cell-centered fields. However, the interpolation is highly sensitive to the hyperpa-

rameter l defining the kernel function and its optimization is subject to further

study.

The recent advent of high-performance computing hardware enables large-scale, multi-physics simulation that provides accurate physical pictures in various fields of study. In order to utilize the high-performance computing system more efficiently, the high-order numerical approximations have become one of the central themes in computational fluid dynamics (CFD) due to their potential in achieving highly accurate predictions in a limited memory capacity.The single-stage or single-step high-order temporal discretizations have shown great promise in delivering high-order temporal accuracy in fast performance. Fundamentally, the single-stage time integrators are based on a Taylor series in the time domain. Although its high performance, the single-stage time integrators are less attractive and less flexible compared to the multi-stage methods due to the complexities in calculating the coefficient of time-Taylor expansion, which usually demands the flux Jacobians and Hessians. This dissertation develops a new single-stage high-order temporal integrator under finite difference discretization. The proposed high-order temporal method is based on the Lax-Wendroff type time discretization, with an algorithmic extension that provides the system independence property. The new approach, called system-free (SF) method, furnishes ease of implementation as well as portability and flexibility of the single-stage time integration method while maintaining the accuracy and stability of the numerical solution.

Across a variety of fields, interpolation algorithms have been used to upsample low

resolution or coarse data fields. In this work, novel Gaussian Process based methods

are employed to solve a variety of upsampling problems. Specifically three

applications are explored: coarse data prolongation in Adaptive Mesh Refinement

(AMR) in the field of Computational Fluid Dynamics, accurate document image

upsampling to enhance Optical Character Recognition (OCR) accuracy, and fast

and accurate Single Image Super Resolution (SISR). For AMR, a new, efficient,

and “3rd order accurate” algorithm called GP-AMR is presented. Next, a novel,

non-zero mean, windowed GP model is generated to upsample low resolution document

images to generate a higher OCR accuracy, when compared to the industry

standard. Finally, a hybrid GP convolutional neural network algorithm is used to

generate a computationally efficient and high quality SISR model.

Next generation computing hardware is expected to deliver large gains

in processing power with less memory resources being the limiting

factor in scalability. High-order methods, compared to low-order

methods, achieve greater solution accuracy on the same grid resolution

through computing higher-order approximations, embodying the necessary

balance between lower memory costs and greater floating point

operations. Until recently polynomial based approaches have dominated

the landscape of high-order spatial discretization. This is in large

part due to their relationship to Taylor expansion, being one of the

most familiar of function approximations. However, the need to fit a

fixed number of parameters can be restrictive, especially on

multidimensional problems where the complexity of polynomial based

numerical methods increases drastically, which compound as the order

of accuracy is increased.

In this dissertation a new framework for designing high-order of accuracy numerical

descriptions for computational fluid dynamics based on the Gaussian

process (GP) family of stochastic functions is developed. Instead of

viewing function approximation as needing to fit a set number of

parameters, GP modelling views the underlying function as belonging to

a space of ``likely'' functions defined by the chosen GP model. This

view of function approximation in terms of a likelihood is exploited

to furnish a robust method for balancing the tension between

high-order approximation and the capturing of discontinuities allowed

in the systems considered.

Systems of hyperbolic conservation laws (HCLs) commonly arise as mathematical descriptors of the natural world, and are particularly ubiquitous in fluid dynamics. These laws appear as complicated and highly nonlinear partial differential equations describing the evolution of fundamental conserved quantities such as mass, momentum, and energy. Solving these equations analytically is entirely intractable for all but the simplest cases, and investigating problems with real world importance falls to numerical approaches more and more frequently. Most HCLs exhibit rich dynamics with complicated smooth flows and discontinuities coexisting, often with shocks arising frominitially smooth data. Designing numerical schemes that can efficiently and accurately represent smooth phenomena, while also remaining robust and reliable in the vicinity of shocks, is very challenging. Finite volume methods are one particularly useful approach to designing such methods as conservation is enforced discretely, and discontinuities can be represented quite naturally. An unfortunate drawback of these methods is that achieving high-order accuracy in multiple space dimensions is difficult. This dissertation overcomes these challenges by developing a kernel-based non-polynomial reconstruction scheme that is manifestly multidimensional. This scheme is first posed as a linear recovery problem in a reproducing kernel Hilbert space. This linear reconstruction method is then cast into a weighted essentially non-oscillatory (WENO) framework so that it may represent both smooth and discontinuous data. This scheme is then incorporated into solvers for the compressible Euler equations, compressible Navier-Stokes equations, and ideal magnetohydrodynamics (MHD) equations. In doing so, a novel set of variables that are more suited to multidimensional reconstruction, dubbed the linearized primitive variables, are introduced. Troubled cell indicators are developed that allow for a more accurate and efficient treatment of smooth solutions in an entirely automatic fashion. Positivity preserving limiters are also incorporated, and allow for the evolution of flows with extremely strong shocks. A highly parallel multi-GPU implementation is provided, and the proposed method is tested against a variety of stringent benchmark problems.

Introduction: Neuroimaging is recommended for patients with seizures to identify intracranial pathology. However, emergency physicians should consider the risks and benefits of neuroimaging in pediatric patients because of their need for sedation and greater sensitivity to radiation than adults. The purpose of this study was to identify associated factors of neuroimaging abnormalities in pediatric patients experiencing their first afebrile seizure.

Methods: This was a retrospective, multicenter study that included children who presented to the emergency departments (ED) of three hospitals due to afebrile seizures between January 2018–December 2020. We excluded children with a history of seizure or acute trauma and those with incomplete medical records. A single protocol was followed in the three EDs for all pediatric patients experiencing their first afebrile seizure. We performed multivariable logistic regression analysis to identify factors associated with neuroimaging abnormalities.

Results: In total, 323 pediatric patients fulfilled the study criteria, and neuroimaging abnormalities were observed in 95 patients (29.4%). Multivariable logistic regression analysis showed that Todd’s paralysis (odds ratio [OR] 3.72, 95% confidence interval [CI] 1.03-13.36; P=0.04), absence of poor oral intake (POI) (OR 0.21, 95% CI 0.05-0.98; P=0.05), lactic acidosis (OR 1.16, 95% CI 1.04-1.30; P=0.01), and higher level of bilirubin (OR 3.33, 95% CI 1.11-9.95; P=0.03) were significantly associated with neuroimaging abnormalities. Based on these results, we constructed a nomogram to predict the probability of brain imaging abnormalities.

Conclusion: Todd’s paralysis, absence of POI, and higher levels of lactic acid and bilirubin were associated factors of neuroimaging abnormalities in pediatric patients with afebrile seizure.