Space-time spectral methods for partial differential equations

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dc.contributor.authorKaur, Avleen
dc.contributor.examiningcommitteeSlevinsky, Richard Mikaëlen_US
dc.contributor.examiningcommitteeWang, Bing-Chenen_US
dc.contributor.examiningcommitteeLiang, Dongen_US
dc.contributor.supervisorLui, Shiu Hong
dc.date.accessioned2022-08-22T15:14:10Z
dc.date.available2022-08-22T15:14:10Z
dc.date.copyright2022-08-21
dc.date.issued2022-08-21
dc.date.submitted2022-08-22T01:05:51Zen_US
dc.degree.disciplineMathematicsen_US
dc.degree.levelDoctor of Philosophy (Ph.D.)en_US
dc.description.abstractSpectral methods for solving partial differential equations (PDEs) depict a high order of convergence, which is exponential when the solution is analytic. However, their applications to time-dependent PDEs typically enforce a finite difference scheme in time. The slower decay of error in time overwhelms the super-algebraic convergence of error in space. A relatively new class of techniques is space-time spectral methods converging spectrally in both space and time. We devise and analyze a space-time spectral method for the Stokes problem. The main objectives of the research are estimating the condition number of the global spectral operators and proving the spectral convergence of this scheme in space and time. Numerical experiments of this scheme verify the theoretical results. Furthermore, we discuss two space-time spectral methods for the Navier-Stokes problem. The discrete systems resulting from classical space-time spectral methods are dense, ill-conditioned, and coupled in all time steps. A new class of spectral methods, called the ultraspherical spectral (US) methods, are applied to time-dependent PDEs, which along with spectral convergence, lead to the resultant discrete systems constituting sparse and well-conditioned matrices. %We present spectral condition number estimates for the heat, Schr\"{o}dinger, and wave equations. Additionally, we join the long tradition of estimating the eigenvalues of a sum of two symmetric matrices, say $P+Q$, in terms of the eigenvalues of $P$ and $Q$. We derive two new lower bounds on $\lambda_{\min}(P +Q)$ in terms of the minimum positive eigenvalues of $P$ and $Q$. The bounds incorporate geometric information by utilizing the Friedrichs angles between certain subspaces. Such estimates lead to new lower bounds on the minimum singular value of some full-rank block matrices in terms of the minimum positive singular value of their subblocks.en_US
dc.description.noteOctober 2022en_US
dc.description.sponsorshipPacific Institute for the Mathematical Sciences and Fields Institute for Research in Mathematical Sciencesen_US
dc.identifier.urihttp://hdl.handle.net/1993/36715
dc.language.isoengen_US
dc.rightsopen accessen_US
dc.subjectspectral methodsen_US
dc.subjectpartial differential equationsen_US
dc.subjecttime-dependent partial differential equationsen_US
dc.subjectminimum singular valueen_US
dc.subjectminimum eigenvalueen_US
dc.subjectspace-time spectral methodsen_US
dc.subjectprincipal anglesen_US
dc.subjectFriedrichs angleen_US
dc.titleSpace-time spectral methods for partial differential equationsen_US
dc.typedoctoral thesisen_US
local.subject.manitobanoen_US
oaire.awardNumberStudent #7784201en_US
oaire.awardTitleUniversity of Manitoba Graduate Fellowshipen_US
oaire.awardURIhttps://umanitoba.ca/graduate-studies/funding-awards-and-financial-aid/university-manitoba-graduate-fellowship-umgfen_US
project.funder.identifierhttps://doi.org/10.13039/100010318en_US
project.funder.nameUniversity of Manitobaen_US
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