Numerical Methods for Differential Equations on Surfaces

Abstract

In this thesis, we present two high-order numerical methods for differential equations on surfaces. In the first part, we introduce an embedding method for solving the Laplace-Beltrami eigenproblem on implicitly defined surfaces. By replacing the surface differential operator in the previous work [1] with a novel formulation and incorporating an extension layer, the method achieves second-order accuracy, surpassing the previous approach, while significantly reducing the computational cost by reducing the size of the computational domain from O(1) to O(∆x). We also introduce the Motion-DNA, which is a key application of our new approach to distinguish the shapes that share identical static spectra. In the second part, we introduce the Spherical Runge-Kutta methods with Richardson Extrapolation (SRKRE) schemes, which is a class of high-order schemes for solving differential equations on the unit sphere. By adapting Richardson-Extrapolation intrinsically to the sphere’s geometry, the method ensures all solutions lie exactly on the unit sphere without additional projection. By combining existing low-order spherical integrators from our previous work [2] with this novel extrapolation mechanism, SRKRE schemes achieve higher-order accuracy and superior performance.

Date of Award	2025
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Shing Yu LEUNG (Supervisor)