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An ultraspherical spectral element method for solving partial differential equations
Thesis (MSc)--Stellenbosch University, 2023.
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| Format: | Thesis |
| Language: | en_ZA en_ZA |
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Stellenbosch : Stellenbosch University
2023
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| _version_ | 1867613799222083584 |
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| access_status_str | Open Access |
| author | Nel, Emma Alida |
| author2 | Hale, Nicholas |
| author_browse | Hale, Nicholas Nel, Emma Alida |
| author_facet | Hale, Nicholas Nel, Emma Alida |
| author_sort | Nel, Emma Alida |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2023. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/128991 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:41:52.972Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/128991 An ultraspherical spectral element method for solving partial differential equations Nel, Emma Alida Hale, Nicholas Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division. Differential equations, Partial Spectral element method Ultraspherical spectral method Chebyshev polynomials Differential equations Thesis (MSc)--Stellenbosch University, 2023. ENGLISH ABSTRACT: We investigate the ultraspherical spectral element method for solving second-order partial differential equations in two dimensions. Moreover, a novel coordinate transformation is introduced to broaden the scope of the method, making it applicable to rectangular domains with circular holes (square donuts), as well as certain types of curved boundaries. The presented method is an integration of two approaches, namely the ultraspherical spectral method and the hierarchical Poincaré–Steklov (HPS) scheme. The ultraspherical method is a Petrov–Galerkin scheme that presents operators in the form of sparse and almost-banded matrices, enabling both stability and computational efficiency. The HPS method is a recursive domain decomposition strategy that enables fast direct solves. It merges solution operators and Dirichlet-to-Neumann operators between subdomains, enforcing continuity of the solution and its derivative across domain boundaries. The fusion of these two methods, combined with a bilinear mapping, results in an accurate discretisation with an explicit direct solve that can be applied to problems on arbitrary polygonal domains with smooth solutions. A major advantage is the reuse of precomputed solution operators facilitated by the HPS scheme, enhancing the efficiency of elliptic solves within implicit and semi-implicit time-steppers. Additionally, the approach is highly parallelisable, allowing for efficient computation time. An implementation of the method is established as a software system, ultraSEM, which employs the HPS method to solve on rectangular and polygonal domains. We extend this implementation to allow for solving on domains with circular cavities. This extension relies on a nonlinear coordinate mapping and proves to work effectively, achieving near machine level precision accuracy. On some simple test problems, we demonstrate geometric convergence for refinement of the polynomial degree and algebraic convergence for domain refinement. Furthermore, we show that execution times scale comparably to those achieved for a rectangular domain. We demonstrate the application of the method on various time-dependent and fluid dynamics examples, including contaminant transport and reaction-diffusion systems, and underscore the practical applicability of the methodology and the new domain. AFRIKAANSE OPSOMMING:Ons ondersoek die ultrasferiese spektrale element metode vir die oplossing van tweedeorde parsiële differensiaalvergelykings in twee dimensies. ’n Nuwe koördinaattransformasie word voorgestel om die omvang van die metode te verbreed en dit van toepassing te maak op reghoekige gebiede met sirkelvormige gate, sowel as sekere tipes gekurfde grense. Die voorgestelde metode is ’n integrasie van twee benaderings, naamlik die ultrasferiese pektrale metode en die hiërargiese Poincaré–Steklov (HPS) skema. Die ultrasferiese metode is ’n Petrov–Galerkin-skema wat operatore skep in die vorm van matrikse wat effektief is om mee te werk en goeie stabiliteit bied. Die HPS-metode is ’n rekursiewe gebied-ontbindingstrategie wat vinnige direkte oplossings moontlik maak. Dit weef oplossings-operatore en Dirichletna- Neumann-operatore tussen subgebiede saam. In die proses word die kontinuïteit van die oplossing en sy afgeleide, oor gebiedsgrense afgedwing. Die samesmelting van hierdie twee strategieë, gekombineer met ’n bilineêre afbeelding, lei tot ’n direkte strategie om akkurate oplossings te genereer vir probleme op veelhoekige gebiede met gladde oplossings. ’n Groot voordeel is die hergebruik van voorafberekende oplossings-operatore wat deur die HPS-skema gefasiliteer word. Dit verbeter veral die doeltreffendheid om elliptiese probleme op te los binne implisiete en semi-implisiete tydstappers. Daarbenewens is dit ook moontlik om die benadering in parallel toe te pas en dus ‘n doeltreffende berekeningstyd moontlik te maak. ’n Reedsbestaande implementering van die metode bestaan wel as ‘n sagtewarestelsel genaamd ultraSEM. Hierdie sagteware gebruik die HPS-metode om oplossings op reghoekige en veelhoekige gebiede te vind. Ons brei hierdie implementering uit om die oplossing op gebiede met sirkelvormige holtes moontlik te maak. Hierdie uitbreiding maak staat op ’n nie-lineêre koördinaat afbeelding en word bewys om effektief te werk. Met ’n paar eenvoudige toetsprobleme word die metode gedemonstreer. Dit vertoon geometriese konvergensie wanneer die polinoomgraad verfyn word en algebraïese konvergensie vir gebiedverfyning. Verder groei die berekeningstyd teen ‘n vergelykbare tempo as dit wat vir ’n vierhoekige gebied behaal word. Ons demonstreer die toepassing van die nuwe metode op verskeie tydafhanklike en vloeidinamika voorbeelde, insluitend kontaminant vervoer en reaksie-diffusie stelsels. Met hierdie voorbeelde word die praktiese toepaslikheid van die metodologie en die nuwe gebied onderstreep. Masters 2023-11-23T12:27:34Z 2024-01-08T18:26:19Z 2023-11-23T12:27:34Z 2024-01-08T18:26:19Z 2023-12 Thesis https://scholar.sun.ac.za/handle/10019.1/128991 en_ZA en_ZA Stellenbosch University vii, 84 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Differential equations, Partial Spectral element method Ultraspherical spectral method Chebyshev polynomials Differential equations Nel, Emma Alida An ultraspherical spectral element method for solving partial differential equations |
| title | An ultraspherical spectral element method for solving partial differential equations |
| title_full | An ultraspherical spectral element method for solving partial differential equations |
| title_fullStr | An ultraspherical spectral element method for solving partial differential equations |
| title_full_unstemmed | An ultraspherical spectral element method for solving partial differential equations |
| title_short | An ultraspherical spectral element method for solving partial differential equations |
| title_sort | ultraspherical spectral element method for solving partial differential equations |
| topic | Differential equations, Partial Spectral element method Ultraspherical spectral method Chebyshev polynomials Differential equations |
| url | https://scholar.sun.ac.za/handle/10019.1/128991 |
| work_keys_str_mv | AT nelemmaalida anultrasphericalspectralelementmethodforsolvingpartialdifferentialequations AT nelemmaalida ultrasphericalspectralelementmethodforsolvingpartialdifferentialequations |