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, 09:59:25pm CET

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Session Overview
MS 31: fast direct solvers for spectral methods
Tuesday, 13/July/2021:
4:00pm - 6:00pm

Session Chair: Per-Gunnar Martinsson
Session Chair: Daniel Frank Fortunato
Virtual location: Zoom 3

Session Abstract

Spectral methods have proven extremely successful at solving differential equa-

tions to high accuracy. However, traditionally such spectral methods result in denser ma-

trices than competing methods, limiting the scale of problems that they can be used for.

In recent years, significant progress on fast direct solvers for spectral methods, enabling

their usage on large scale problems, has been made on two fronts: domain decomposition

methods and structured operators in coefficient space. This minisymposia brings together

speakers in both research directions.

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4:00pm - 4:30pm

A stable spectral solver for hyperbolic equations

Ke Chen1, Daniel Appelo2, Tracy Babb3, Per-Gunnar Martinsson1

1University of Texas at Austin, United States of America; 2Michigan State University; 3University of Colorado Boulder

In this paper, we present a high order stable spectral solver for general nonlinear hyperbolic equations. We use additive Runge-Kutta methods for time-stepping, which integrate the linear stiff terms by an explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is computed via the recently developed Hierarchical Poincare-Steklov (HPS) method. The HPS method is a fast direct solver for elliptic equations which decomposes the space domain into a hierarchical tree of subdomains and builds spectra collocation solvers locally on the subdomains. These ideas are naturally combined in our method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures same coefficient matrix in the implicit solve of HPS for all time stages and can be efficiently reused. Stability of the proposed method is proved for first order in time and high order in space. We verify the stability and accuracy of the proposed method in several numerical examples.

4:30pm - 5:00pm

The ultraspherical spectral element method

Daniel Fortunato1, Nicholas Hale2, Alex Townsend3

1Flatiron Institute; 2Stellenbosch University; 3Cornell University

We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincare-Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured quadrilateral or triangular meshes. Properties of the ultraspherical spectral method lead to almost banded linear systems, allowing the element method to be competitive in the high-polynomial regime. The hierarchical Poincare-Steklov scheme enables precomputed solution operators to be reused, allowing for fast elliptic solves in implicit and semi-implicit time-steppers. The resulting spectral element method achieves an overall computational complexity of $\mathcal{O}(p^4/h^3)$ for mesh size $h$ and polynomial order $p$, enabling $hp$-adaptivity to be efficiently performed. We develop an open-source software system, \ultraSEM, for flexible, user-friendly spectral element computations in MATLAB. Extensions to 3D are briefly discussed.

5:00pm - 5:30pm

Time and Frequency Domain Methods for the Wave Equation with High Order and Tame CFL

Daniel Appelo

Michigan State University, United States of America

A new iterative method, the WaveHoltz iteration, for solution of the Helmholtz equation is presented. WaveHoltz is a fixed-point iteration that filters the solution to the solution of wave equation with time-periodic forcing and boundary data. The WaveHoltz iteration corresponds to a linear and coercive operator which, after discretization, can be recast as a positive definite linear system of equations. The solution to this system of equations approximates the Helmholtz solution and can be accelerated by Krylov subspace techniques.

The WaveHoltz method can be driven by any wave equation solver. Preferably such solvers should be able to march with large timesteps bounded only by the (physical) finite speed of propagation. Here we pair the WaveHoltz method with Hermite methods and a new central discontinuous Galerkin method, both able to march with CFL conditions independent of the order of approximation.

5:30pm - 6:00pm

Ultraspherical spectral method for the computation of power law equilibrium measures

Timon S. Gutleb1, José A. Carrillo2, Sheehan Olver1

1Imperial College London, United Kingdom; 2University of Oxford, United Kingdom

Equilibrium measure problems naturally appear as the continuous limit of particle swarms in which particle behavior may be modeled via attractive and repulsive forces. We introduce a banded spectral method using weighted ultraspherical polynomials which allows the efficient computation of equilibrium measures for power law kernels. The justification of the method involves the derivation of recurrence relationships of these polynomial bases for Riemann-Liouville integrals, Riesz potentials and the fractional Laplacian. We present numerical experiments which agree with special case analytic results as well as independent particle swarm simulations. Finally, we sketch how future research will extend these methods to higher dimensional problems.

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