Gutie, J. (SciCAM) – SORh: Hyperbolic Relaxation Methods For Elliptic Problems In Computational Fluid Dynamics

This thesis explores iterative methods for solving elliptic partial differential equations (PDEs), which are used in computational fluid dynamics (CFD) to model a wide range of physical phenomena. The primary application of interest here is self-gravity, modeled by Poisson’s equation. Although many numerical approaches exist, including direct matrix inversion, FFT-based methods, and classical iterative methods such as Jacobi and Gauss-Seidel, these approaches involve tradeoffs in computational cost, scalability, implementation complexity, and adaptability to changing boundary conditions and problem configurations.

Therefore, we introduce SORh, a simple and efficient relaxation method derived from a hyperbolic reformulation of Poisson’s equation. SORh generalizes classical successive over-relaxation (SOR) by providing independent control of residual relaxation and the directional propagation of Gauss–Seidel corrections. We present formulations of SORh in one and two spatial dimensions and investigate its stability, accuracy, and computational performance through analytical derivations and numerical comparisons with established relaxation methods. The results identify favorable SORh formulations, clarify their relationships to classical relaxation methods, and demonstrate improved convergence on selected test problems. Finally, we demonstrate applications of SORh to astrophysical self-gravity simulations in the FLASH code and to magnetohydrodynamic (MHD) divergence cleaning.

Event Host: Jonathan Guite, M.S. Candidate, Scientific Computing & Applied Mathematics 

Advisor: Dongwook Lee

Zoom: https://ucsc.zoom.us/j/92153750104?pwd=ZdLiDZeLqOAlVNX9C4bCloKno9tAeB.1

Passcode: 769232