In partial differential equations (PDEs), conserved quantities like mass and momentum are fundamental to understanding the behavior of the described physical
systems. The preservation of conserved quantities is essential when using numerical
schemes to approximate solutions of corresponding PDEs. If the discrete solutions
obtained through these schemes fail to preserve the conserved quantities, they may
be physically meaningless and unreliable.
Previous approaches focused on checking conservation in PDEs and numerical
schemes, but they did not give adequate attention to systematically handling parameters. This is a crucial aspect because many PDEs and numerical schemes have parameters that need to be dealt with systematically. Here, we investigate if the discrete
analog of a conserved quantity is preserved under the solution induced by a parametric finite difference method. In this thesis, we modify and enhance a pre-existing
algorithm to effectively and reliably deduce conserved quantities in the context of
parametric schemes, using the concept of comprehensive Groebner systems.
The main contribution of this work is the development of a versatile algorithm
capable of handling various parametric explicit and implicit schemes, higher-order
derivatives, and multiple spatial dimensions. The algorithm’s effectiveness and efficiency are demonstrated through examples and applications. In particular, we illustrate the process of selecting an appropriate numerical scheme among a family of
potential discretization for a given PDE.
| Identifer | oai:union.ndltd.org:kaust.edu.sa/oai:repository.kaust.edu.sa:10754/693163 |
| Date | 05 1900 |
| Creators | Majrashi, Bashayer |
| Contributors | Gomes, Diogo A., Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division, Parsani, Matteo, Dominik L, Michele |
| Source Sets | King Abdullah University of Science and Technology |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Relation | N/A |
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