Dissertation
Wiggle-Free Leapfrog Scheme and an Application of the Operator Splitting Method for Tracking Contaminant Transport
Washington State University
Doctor of Philosophy (PhD), Washington State University
2023
DOI:
https://doi.org/10.7273/000005291
Abstract
This dissertation presents the construction of some innovative numerical methods that approximate certain partial differential equations of practical interest. Partial differential equations are found to be useful in describing physical phenomena in the real-world and model problems encountered in engineering, technology and the life sciences. The numerical methods proposed and constructed in this thesis are motivated by the idea of simplifying some complex computational approaches. We first study the numerical approximation of the advection equation which has been of interest to the weather research community and other fluid flow researchers, using the leapfrog scheme and thereby investigate the unwanted dispersive wiggles that arise from the leapfrog scheme. We develop a computationally efficient numerical method that simply eliminates the dispersive wiggles while keeping the solution profile intact. We extend our approach and implement it to enhance the Robert-Asselin (RA) filter that is widely used in weather modeling for controlling leapfrog scheme’s wiggles. The numerical method that we implement successfully controls the wiggles and holds the amplitude of the solution profile. We further extend our work by introducing a dispersion term to the advection problem and study the advection – dispersion equation (ADE). The ADE is widely used to describe the physical phenomena encountered in various engineering problems and life processes. We study the ADE to understand the contaminant transport as it describes the advection (movement) and dispersion (spread) of a contaminant plume. We present a new numerical discretization using the operator splitting method to solve the ADE forward in time and after successful validation of the method, we use it along with variable transformation to solve the ADE backward in time. We present the numerical simulations to validate our method and its effectiveness of the numerical approximation to capture the solution profiles. We, also, construct a new discretization to approximate an ADE that involves the sorption process along with advection and dispersion. For every numerical method that is developed in the following chapters, we determine its stability and present any conditions that determine how any associated parameters are chosen.
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Details
- Title
- Wiggle-Free Leapfrog Scheme and an Application of the Operator Splitting Method for Tracking Contaminant Transport
- Creators
- Priyanka Rao
- Contributors
- Valipuram S Manoranjan (Advisor)Hong-Ming Yin (Committee Member)Sergey Lapin (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 108
- Identifiers
- 99901019232301842
- Language
- English
- Resource Type
- Dissertation