Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation-by-parts operators

Pieter D. Boom; David W. Zingg

Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation-by-parts operators

Pieter D. Boom, David W. Zingg

University of Toronto

Research output: Chapter in Book/Conference proceeding › Conference contribution › peer-review

Abstract

This paper summarizes several new developments in the theory of high-order implicit Runge-Kutta (RK) methods based on generalized summation-by-parts (GSBP) operators. The theory is applied to the construction of several known and novel Runge-Kutta schemes. This includes the well-known families of fully-implicit Radau IA/IIA and Lobatto IIIC Runge-Kutta methods. In addition, a novel family of GSBP-RK schemes based on Gauss quadrature rules is presented along with a few optimized diagonally-implicit GSBP-RK schemes. The novel schemes are all L-stable and algebraically stable. The stability and relative efficiency of the schemes is investigated with numerical simulation of the linear convection equation with both time-independent and time-dependent convection velocities. The numerical comparison includes a few popular non-GSBP Runge-Kutta time-marching methods for reference.

Original language	English
Title of host publication	22nd AIAA Computational Fluid Dynamics Conference
Publisher	American Institute of Aeronautics and Astronautics
ISBN (Print)	9781624103667
Publication status	Published - 1 Jan 2015
Event	22nd AIAA Computational Fluid Dynamics Conference, 2015 - Dallas, United States Duration: 22 Jun 2015 → 26 Jun 2015

Conference

Conference	22nd AIAA Computational Fluid Dynamics Conference, 2015
Country/Territory	United States
City	Dallas
Period	22/06/15 → 26/06/15

Cite this

@inproceedings{3170c94d33fa47a0b7e1f58944586915,

title = "Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation-by-parts operators",

abstract = "This paper summarizes several new developments in the theory of high-order implicit Runge-Kutta (RK) methods based on generalized summation-by-parts (GSBP) operators. The theory is applied to the construction of several known and novel Runge-Kutta schemes. This includes the well-known families of fully-implicit Radau IA/IIA and Lobatto IIIC Runge-Kutta methods. In addition, a novel family of GSBP-RK schemes based on Gauss quadrature rules is presented along with a few optimized diagonally-implicit GSBP-RK schemes. The novel schemes are all L-stable and algebraically stable. The stability and relative efficiency of the schemes is investigated with numerical simulation of the linear convection equation with both time-independent and time-dependent convection velocities. The numerical comparison includes a few popular non-GSBP Runge-Kutta time-marching methods for reference.",

author = "Boom, {Pieter D.} and Zingg, {David W.}",

year = "2015",

month = jan,

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language = "English",

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booktitle = "22nd AIAA Computational Fluid Dynamics Conference",

publisher = "American Institute of Aeronautics and Astronautics ",

address = "United States",

note = "22nd AIAA Computational Fluid Dynamics Conference, 2015 ; Conference date: 22-06-2015 Through 26-06-2015",

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TY - GEN

T1 - Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation-by-parts operators

AU - Boom, Pieter D.

AU - Zingg, David W.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - This paper summarizes several new developments in the theory of high-order implicit Runge-Kutta (RK) methods based on generalized summation-by-parts (GSBP) operators. The theory is applied to the construction of several known and novel Runge-Kutta schemes. This includes the well-known families of fully-implicit Radau IA/IIA and Lobatto IIIC Runge-Kutta methods. In addition, a novel family of GSBP-RK schemes based on Gauss quadrature rules is presented along with a few optimized diagonally-implicit GSBP-RK schemes. The novel schemes are all L-stable and algebraically stable. The stability and relative efficiency of the schemes is investigated with numerical simulation of the linear convection equation with both time-independent and time-dependent convection velocities. The numerical comparison includes a few popular non-GSBP Runge-Kutta time-marching methods for reference.

AB - This paper summarizes several new developments in the theory of high-order implicit Runge-Kutta (RK) methods based on generalized summation-by-parts (GSBP) operators. The theory is applied to the construction of several known and novel Runge-Kutta schemes. This includes the well-known families of fully-implicit Radau IA/IIA and Lobatto IIIC Runge-Kutta methods. In addition, a novel family of GSBP-RK schemes based on Gauss quadrature rules is presented along with a few optimized diagonally-implicit GSBP-RK schemes. The novel schemes are all L-stable and algebraically stable. The stability and relative efficiency of the schemes is investigated with numerical simulation of the linear convection equation with both time-independent and time-dependent convection velocities. The numerical comparison includes a few popular non-GSBP Runge-Kutta time-marching methods for reference.

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M3 - Conference contribution

SN - 9781624103667

BT - 22nd AIAA Computational Fluid Dynamics Conference

PB - American Institute of Aeronautics and Astronautics

T2 - 22nd AIAA Computational Fluid Dynamics Conference, 2015

Y2 - 22 June 2015 through 26 June 2015

ER -

Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation-by-parts operators

Abstract

Conference

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