In this work, we develop numerical methods for conservation laws that explore statistical, structure-preserving, and machine-learning-based approaches, each built on top of traditional numerical solvers. First, we develop a general Gaussian-process-based “recipe’’ for constructing high-order linear operators such as interpolation, reconstruction, and derivative approximations. Building on this recipe, we derive a kernel-agnostic convergence theory for GP-based operators that interprets them as generalized finite-difference schemes, defines an effective order-of-accuracy proxy that captures non-ideal truncation-error structure, and uses this metric to select stencil geometries and kernel hyperparameters analytically. We then introduce a new second-order kernel, Discontinuous Arcsin (DAS), that is stationary and prevents oscillations. DAS is integrated into a shock-capturing framework called the Multidimensional Optimal Order Detection (MOOD) method and shows an increase in efficiency by admitting less first order cascades. Next, we address the long-standing problem of spurious pressure oscillations in compressible multi-component and real-fluid simulations by introducing a fully conservative pressure-equilibrium-preserving scheme and a high-order fully conservative approximate variant that apply to arbitrary equations of state. Unlike existing approaches, these methods avoid non-conservative updates or EOS-specific constructions, and on smooth interface advection tests with ideal-gas, stiffened-gas, and van der Waals fluids they reduce spurious pressure oscillations by orders of magnitude relative to current schemes. We then propose a hybrid numerical–machine learning framework for mixed hyperbolic–parabolic systems in which only the diffusive contribution is learned while the hyperbolic fluxes are advanced with standard shock-capturing methods, enabling timesteps at a hyperbolic CFL. Within this framework, we compare several neural architectures and loss designs on viscous Burgers tests and on the one-dimensional Euler equations with heat conduction, showing that U-shaped neural operators combined with multi-step and TVD-style regularization improve long-time stability and spectral behavior, and we analyze the resulting coupled schemes via eigenvalue-based stability diagnostics. Finally, we apply high-order, shock-capturing finite-difference methods within NASA’s Launch Ascent and Vehicle Aerodynamics (LAVA) framework to quantify acoustic and pressure loads on the Artemis Mobile Launcher, including multiphase simulations of water-suppression systems and comparisons to flight data that inform hardware design for future missions. Collectively, this work offers a set of targeted advances in kernel-based numerical operators, conservative schemes and learning-augmented solvers each aimed at improving accuracy, stability, or efficiency in complex multiphysics flow simulation.
Event Host: Chris DeGrendele, Ph.D. Candidate, Applied Mathematics
Advisor: Dongwook Lee
Zoom- https://ucsc.zoom.us/j/96308438100?pwd=9El4idgPoaVnAd9m8M6As6uaSbcojp.1
Passcode- 123456
