UC Santa Cruz

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.