Publication Date

5-2025

Date of Final Oral Examination (Defense)

2-28-2025

Type of Culminating Activity

Dissertation

Degree Title

Doctor of Philosophy in Computing

Department

Computer Science

Supervisory Committee Chair

Grady B. Wright, Ph.D.

Supervisory Committee Member

Donna Calhoun, Ph.D.

Supervisory Committee Member

Jodi Mead, Ph.D.

Supervisory Committee Member

Peter Bosler, Ph.D.

Abstract

Constructing high-order accurate numerical methods for the polar and spherical geometry is an essential task in many areas of computational science. This dissertation consists of three projects related to the development and application of the Double Fourier Sphere (DFS) method, which is particularly well-suited for approximation of tensor product gridded data on the sphere and disk. These grids are commonly used in weather or climate forecasting, optics, and astrophysics. However they present issues for numerical methods because of artificial boundaries that are introduced by spherical and polar coordinates. The DFS method transforms a sphere into a doubly periodic domain without artificial boundaries at the poles and a disk into a domain without an artificial boundary at the origin.

In the first project we use the DFS method to develop barycentric interpolation formulas for gridded data on the sphere and disk that bypasses the issues associated with high-order approximations on these geometries. We demonstrate how to exploit symmetries associated with the DFS method to derive bivariate trigonometric interpolation formulas for the sphere and bivariate trigonometric-polynomial interpolation formulas for the disks. These barycentric formulas are exponentially accurate for approximating smooth functions. We also show how the formulas can be accelerated through the application of the Nonuniform Fast Fourier Transform (NUFFT) without compromising accuracy.

In the second project, we conduct the first numerical investigation of spectrally accurate interpolation in Semi-implicit semi-Lagrangian (SISL) schemes for the shallow water equations (SWE). SISL methods are commonly used for the SWE because they allow for larger time steps than those permitted by the Courant-Friedrichs-Lewy (CFL) stability condition in Eulerian schemes. Operational SISL schemes routinely employ spectrally accurate spatial discretizations, such as spherical harmonics or the double Fourier sphere (DFS) method, for computing horizontal derivatives of the prognostic variables. This creates a mismatch in numerical accuracy, making the use of low-order interpolation less clearly justified. The second project addresses this by incorporating DFS-based spectral interpolation from the first project into SISL schemes while maintaining original computational complexity. Using several standard SWE test cases, we evaluate the accuracy, conservation, and numerical diffusion of the new model, particularly over long integration times. Compared to an equivalent SISL model with low-order interpolation, the new model achieves higher accuracy, improved mass and energy conservation, and reduced numerical diffusion, demonstrating the potential benefits of incorporating spectrally accurate interpolation into SISL schemes.

In the final project we exploit the DFS method to develop a high-order hybrid Lagrangian-Eulerian method for fluid dynamics on the rotating sphere. Vortex-based methods for 2D incompressible fluid on a rotating sphere typically require computing fluid velocity by applying the Biot-Savart law. The discretization of the Biot-Savart integral results in an N-body problem, which if solved by direct methods, has a computational cost that scales as O(N^2), where N is the number of particles. Additionally, discretization of this integral requires the particles be arranged in specific ways that are impossible to maintain during the simulation beyond the initial time-step. To bypass this problem, it is common to redistribute the particles after every few time-steps, which introduces additional cost. In this work we developed a cost effective alternative approach that addresses these challenges. In this approach we compute the velocity at each time-step by 1) interpolating the vorticity to a fixed tensor-product grid, 2) solving a Poisson equation for the stream function on the grid, 3) computing velocity on the grid from the stream function, 4) interpolating the velocity back to the particles. All the grid-based discretization are done efficiently, and accurately using DFS method based algorithms. We interpolate the vorticity from the particles to the Eulerian grid using the generalized moving least squares (GMLS) interpolation. And use DFS based interpolation methods developed in the first project to interpolate velocity from the grid to the particles. We use standard examples to show that the proposed scheme is efficient, accurate, and stable.

Comments

Michael Chiwere, ORCID: 0009-0008-7510-9197

DOI

https://doi.org/10.18122/td.2400.boisestate

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Available for download on Saturday, May 01, 2027

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