# Spectral deferred corrections with fast-wave slow-wave splitting

@article{Ruprecht2016SpectralDC, title={Spectral deferred corrections with fast-wave slow-wave splitting}, author={Daniel Ruprecht and Robert Speck}, journal={ArXiv}, year={2016}, volume={abs/1602.01626} }

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves (``fast-wave slow-wave problem''). We show that for a scalar test problem with two imaginary eigenvalues $i \lambda_{\text{f}}$, $i \lambda_{\text{s}}$, having $\Delta t ( | \lambda_{\text{f}} | + | \lambda_{\text{s}} | ) < 1$ is sufficient for the fast-wave slow-wave SDC (fwsw-SDC) iteration to converge and that in the limit of…

## 24 Citations

Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

- MathematicsJ. Comput. Phys.
- 2019

On the convergence of spectral deferred correction methods

- Computer ScienceCommunications in Applied Mathematics and Computational Science
- 2019

This work analyzes the convergence properties of the Spectral Deferred Correction method and indicates that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE.

Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

- Computer ScienceJ. Sci. Comput.
- 2018

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial…

Parallelizing spectral deferred corrections across the method

- Computer ScienceComput. Vis. Sci.
- 2018

Two strategies to enable “parallelization across the method” for spectral deferred corrections (SDC) are presented and the first aims at finding parallel preconditioners for the Picard iteration and the second diagonalizes the quadrature matrix of the collocation problem directly.

pySDC - Prototyping spectral deferred corrections

- Computer ScienceArXiv
- 2018

This paper presents the Python framework pySDC for solving collocation problems with spectral deferred correction methods (SDC) and their time-parallel variant PFASST, the parallel full approximation scheme in space and time, and describes the structure of the code by taking two different perspectives: the user and developer's perspective.

Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

- Computer ScienceJ. Comput. Phys.
- 2020

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

- Computer ScienceNumer. Linear Algebra Appl.
- 2018

This paper uses this multigrid formulation and PFASST's iteration matrix to show that, in the nonstiff and stiff limit,PFASST indeed is a convergent iterative method, and provides upper bounds for the spectral radius of the iteration matrix.

The Parallel Full Approximation Scheme in Space and Time for a Parabolic Finite Element Problem

- Computer ScienceArXiv
- 2021

This paper derives thePFASST algorithm with mass matrices and appropriate prolongation and restriction operators and shows numerically that PFASST can, after some initial iterations, gain two orders of accuracy per iteration.

Using Performance Analysis Tools for a Parallel-in-Time Integrator

- Computer ScienceSpringer Proceedings in Mathematics & Statistics
- 2021

The application of performance analysis tools to the PFASST implementation pySDC is demonstrated and it is hoped that the results and in particular the way they obtained are a blueprint for other time-parallel codes.

A unified analysis framework for iterative parallel-in-time algorithms

- Computer ScienceArXiv
- 2022

It is proved that all four Parareal, PFASST, MGRIT and STMG methods eventually converge super-linearly and compare them directly numerically.

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