# 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…

## Figures, Tables, and Topics from this paper

## 23 Citations

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

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2019

This work proposes a high-order, implicit–explicit numerical scheme that combines Multi-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics (SH) transform to solve the wave-propagation problems arising from the shallow-water equations on the rotating sphere.

On the convergence of spectral deferred correction methods

- MathematicsCommunications in Applied Mathematics and Computational Science
- 2019

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this…

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

- Mathematics, 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…

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

- Mathematics, Computer ScienceNumer. Linear Algebra Appl.
- 2018

This paper will use this multigrid formulation and in particular PFASST's iteration matrix to show that in the non-stiff as well as in the stiff limitPFASST indeed is a convergent iterative method.

Parallelizing spectral deferred corrections across the method

- Computer Science, MathematicsComput. 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

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2020

A multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time with a spatial discretization based on Spherical Harmonics (SH) is proposed, designed to coarsen and interpolate the problem in space using the spectral basis of the SH that is key for the accuracy and efficiency of the approach.

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

- Computer Science, MathematicsArXiv
- 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.

PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method

- Computer Science, MathematicsArXiv
- 2019

This work couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator, which works more efficiently than the classical variants ofPFASST and can use more processors than time-steps.

## References

SHOWING 1-10 OF 54 REFERENCES

Implicit-Explicit Runge-Kutta Methods for Fast-Slow Wave Problems

- Mathematics
- 2013

AbstractLinear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need…

Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems

- Mathematics
- 2012

AbstractImplicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave–slow-wave problems in which the fast wave has relatively low…

A scale-selective multilevel method for long-wave linear acoustics

- Computer Science
- 2011

A new method for the numerical integration of the equations for one-dimensional linear acoustics with large time steps is presented. While it is capable of computing the “slaved” dynamics of…

Accelerating the convergence of spectral deferred correction methods

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2006

It is shown that for linear problems, the iterations in the SDC algorithm are equivalent to constructing a preconditioned Neumann series expansion for the solution of the standard collocation discretization of the ODE.

Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations

- Mathematics
- 2014

AbstractThe compressible Euler equations exhibit wave phenomena on different scales. A suitable spatial discretization results in partitioned ordinary differential equations where fast and slow modes…

Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics

- Mathematics
- 2004

In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales, rendering the reaction part of the…

Spectral Deferred Correction Methods for Ordinary Differential Equations

- Mathematics
- 2000

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a…

High-order multi-implicit spectral deferred correction methods for problems of reactive flow

- Mathematics
- 2003

Models for reacting flow are typically based on advection-diffusion-reaction (A-D-R) partial differential equations. Many practical cases correspond to situations where the relevant time scales…

Linear Stability Analysis of Runge–Kutta-Based Partial Time-Splitting Schemes for the Euler Equations

- Mathematics
- 2010

Abstract For atmospheric simulation models with resolutions from about 10 km to the subkilometer cloud-resolving scale, the complete nonhydrostatic compressible Euler equations are often used. An…

A multi-level spectral deferred correction method

- Computer Science, Mathematics
- 2015

This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels.