Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 30th Nov 2022, 08:53:55pm CET

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Session Overview
MS 9: space-time spectral methods and their fast solvers
Wednesday, 14/July/2021:
4:10pm - 6:10pm

Session Chair: Shaun Lui
Virtual location: Zoom 1

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4:10pm - 4:40pm

Space-time Spectral Methods for PDEs

Sarah Nataj, Shaun Lui

University Of Manitoba, Canada

Spectral methods solve elliptic PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time-dependent PDEs, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Space-time spectral methods are new methods that apply spectral discretization in both space and time. We report on exponential convergence of such methods for the heat, Schrodinger, Airy, beam, and wave equations. The condition number of the methods are estimated. Numerical experiments show that the method also converges spectrally for many common nonlinear PDEs: Allen--Cahn, Burgers, Cahn--Hilliard, KdV, Sine Gordon, Kuramoto—Shivashinsky, and nonlinear reaction diffusions equations. This is joint work with Dr. S.H. Lui.

4:40pm - 5:10pm

A space-time spectral method for the Stokes problem

Avleen Kaur, S. H. Lui

University of Manitoba, Canada

In this work, we consider the Stokes equations in steady and unsteady states, along with Dirichlet boundary conditions and an initial condition in the latter case. We impose the $\mathbb{P}_N-\mathbb{P}_{N-2}$ spectral Galerkin scheme in space by using a recombined Legendre polynomial basis resulting in exponential convergence in space. For the unsteady state, we implement spectral collocation in time, thus giving exponential convergence in both space and time. The global spectral operator for both schemes is a saddle point matrix. We prove the 2-norm estimates for every block of the two operator matrices, and proceed to show that the condition number for the global spectral operator for the steady-state scheme is $\mathcal{O}(N^4)$, where $N$ is the number of spectral modes in each direction. We also have results on the condition number of the unsteady-state scheme. Numerical results of this scheme applied to the unsteady Navier-Stokes problem will also be shown.

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