High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

Gottlieb, Sigal; Grant, Zachary J.; Hu, Jingwei; Shu, Ruiwen

doi:10.1137/21m1403175

Title: High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

Abstract

In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.

Authors:

^[1];

^[2]; Hu, Jingwei ^[3]; Shu, Ruiwen ^[4]

Univ. of Massachusetts, Dartmouth, MA (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Purdue Univ., West Lafayette, IN (United States)
Univ. of Maryland, College Park, MD (United States)

Publication Date:: Tue Feb 15 00:00:00 EST 2022

Research Org.:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE; US Air Force Office of Scientific Research (AFOSR); National Science Foundation (NSF)

OSTI Identifier:: 1867763

Grant/Contract Number:: AC05-00OR22725; FA9550-18-1-0383; DMS-1654152

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Numerical Analysis

Additional Journal Information:: Journal Volume: 60; Journal Issue: 1; Journal ID: ISSN 0036-1429

Publisher:: Society for Industrial and Applied Mathematics (SIAM)

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; asymptotic preserving; strong stability preserving; implicit-explicit scheme; multi-derivative; positivity preserving; high order

Citation Formats


                    Gottlieb, Sigal, Grant, Zachary J., Hu, Jingwei, and Shu, Ruiwen. High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties.  United States: N. p., 2022. 
Web.  doi:10.1137/21m1403175.

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                    Gottlieb, Sigal, Grant, Zachary J., Hu, Jingwei, & Shu, Ruiwen. High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties.  United States.  https://doi.org/10.1137/21m1403175

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                    Gottlieb, Sigal, Grant, Zachary J., Hu, Jingwei, and Shu, Ruiwen. Tue .  
"High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties".  United States.  https://doi.org/10.1137/21m1403175.  https://www.osti.gov/servlets/purl/1867763.

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@article{osti_1867763,

  title        = {High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties},

  author       = {Gottlieb, Sigal and Grant, Zachary J. and Hu, Jingwei and Shu, Ruiwen},

  abstractNote = {In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.},

  doi          = {10.1137/21m1403175},

  journal      = {SIAM Journal on Numerical Analysis},

  number       = 1,

  volume       = 60,

  place        = {United States},

  year         = {Tue Feb 15 00:00:00 EST 2022},

  month        = {Tue Feb 15 00:00:00 EST 2022}

}

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Works referenced in this record:

Hyperbolic conservation laws with stiff relaxation terms and entropy
journal, June 1994

Chen, Gui-Qiang; Levermore, C. David; Liu, Tai-Ping
Communications on Pure and Applied Mathematics, Vol. 47, Issue 6
DOI: 10.1002/cpa.3160470602

Asymptotic Preserving Implicit-Explicit Runge--Kutta Methods for Nonlinear Kinetic Equations
journal, January 2013

Dimarco, Giacomo; Pareschi, Lorenzo
SIAM Journal on Numerical Analysis, Vol. 51, Issue 2
DOI: 10.1137/12087606X

On maximum-principle-satisfying high order schemes for scalar conservation laws
journal, May 2010

Zhang, Xiangxiong; Shu, Chi-Wang
Journal of Computational Physics, Vol. 229, Issue 9
DOI: 10.1016/j.jcp.2009.12.030

Strong Stability-Preserving High-Order Time Discretization Methods
journal, January 2001

Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan
SIAM Review, Vol. 43, Issue 1
DOI: 10.1137/S003614450036757X

Total-Variation-Diminishing Time Discretizations
journal, November 1988

Shu, Chi-Wang
SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 6
DOI: 10.1137/0909073

Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order
journal, September 2017

Conde, Sidafa; Gottlieb, Sigal; Grant, Zachary J.
Journal of Scientific Computing, Vol. 73, Issue 2-3
DOI: 10.1007/s10915-017-0560-2

Efficient implementation of essentially non-oscillatory shock-capturing schemes
journal, August 1988

Shu, Chi-Wang; Osher, Stanley
Journal of Computational Physics, Vol. 77, Issue 2
DOI: 10.1016/0021-9991(88)90177-5

Shock Structure in a Simple Discrete Velocity Gas
journal, January 1964

Broadwell, James E.
Physics of Fluids, Vol. 7, Issue 8
DOI: 10.1063/1.1711368

Implicit-Explicit Linear Multistep Methods for Stiff Kinetic Equations
journal, January 2017

Dimarco, Giacomo; Pareschi, Lorenzo
SIAM Journal on Numerical Analysis, Vol. 55, Issue 2
DOI: 10.1137/16M1063824

A Second-Order Asymptotic-Preserving and Positivity-Preserving Exponential Runge--Kutta Method for a Class of Stiff Kinetic Equations
journal, January 2019

Hu, Jingwei; Shu, Ruiwen
Multiscale Modeling & Simulation, Vol. 17, Issue 4
DOI: 10.1137/18M1226774

Step Sizes for Strong Stability Preservation with Downwind-Biased Operators
journal, January 2011

Ketcheson, David I.
SIAM Journal on Numerical Analysis, Vol. 49, Issue 4
DOI: 10.1137/100818674

Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation
journal, January 2018

Hu, Jingwei; Shu, Ruiwen; Zhang, Xiangxiong
SIAM Journal on Numerical Analysis, Vol. 56, Issue 2
DOI: 10.1137/17M1144362

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
journal, May 1954

Bhatnagar, P. L.; Gross, E. P.; Krook, M.
Physical Review, Vol. 94, Issue 3
DOI: 10.1103/PhysRev.94.511

Strong Stability for Additive Runge–Kutta Methods
journal, January 2006

Higueras, Inmaculada
SIAM Journal on Numerical Analysis, Vol. 44, Issue 4
DOI: 10.1137/040612968

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model
journal, March 2017

Huang, Juntao; Shu, Chi-Wang
Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 03
DOI: 10.1142/S0218202517500099

Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit
journal, January 2013

Boscarino, S.; Pareschi, L.; Russo, G.
SIAM Journal on Scientific Computing, Vol. 35, Issue 1
DOI: 10.1137/110842855

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes
journal, January 2016

Christlieb, Andrew J.; Gottlieb, Sigal; Grant, Zachary
Journal of Scientific Computing, Vol. 68, Issue 3
DOI: 10.1007/s10915-016-0164-2

Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations
journal, January 1999

Jin, Shi
SIAM Journal on Scientific Computing, Vol. 21, Issue 2
DOI: 10.1137/S1064827598334599

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Title: High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

Abstract

Citation Formats

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Efficient implementation of essentially non-oscillatory shock-capturing schemes journal, August 1988

Shock Structure in a Simple Discrete Velocity Gas journal, January 1964

Implicit-Explicit Linear Multistep Methods for Stiff Kinetic Equations journal, January 2017

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Step Sizes for Strong Stability Preservation with Downwind-Biased Operators journal, January 2011

Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation journal, January 2018

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems journal, May 1954

Strong Stability for Additive Runge–Kutta Methods journal, January 2006

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Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations journal, January 1999

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