Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

Conde, Sidafa; Gottlieb, Sigal; Grant, Zachary J.; Shadid, John N.

doi:10.1007/s10915-017-0560-2

Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

Journal Article · Mon Sep 18 00:00:00 EDT 2017 · Journal of Scientific Computing

DOI:https://doi.org/10.1007/s10915-017-0560-2· OSTI ID:1574706

Conde, Sidafa ^[1]; Gottlieb, Sigal ^[2]; Grant, Zachary J. ^[2]; Shadid, John N. ^[3]

Univ. of Massachusetts, North Dartmouth, MA (United States). Dept. of Mathematics; Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Computational Mathematics Dept.
Univ. of Massachusetts, North Dartmouth, MA (United States). Dept. of Mathematics
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Computational Mathematics Dept.; Univ. of New Mexico, Albuquerque, NM (United States). Dept. of Mathematics and Statistics

Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. The search for high order strong stability preserving time-stepping methods with high order and large allowable time-step has been an active area of research. It is known that implicit SSP Runge–Kutta methods exist only up to sixth order; however, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge–Kutta methods of any linear order. In the current work we find implicit SSP Runge–Kutta methods with high linear order p_lin ≤ 9 and nonlinear orders p = 2,3,4, that are optimal in terms of allowable SSP time-step. Next, we formulate a novel optimization problem for implicit–explicit (IMEX) SSP Runge–Kutta methods and find optimized IMEX SSP Runge–Kutta pairs that have high linear order p_lin ≤ 7 and nonlinear orders up to p = 4. Here, we also find implicit methods with large linear stability regions that pair with known explicit SSP Runge–Kutta methods. These methods are then tested on sample problems to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1574706

Report Number(s):: SAND--2019-8924J; 678045

Journal Information:: Journal of Scientific Computing, Journal Name: Journal of Scientific Computing Journal Issue: 2-3 Vol. 73; ISSN 0885-7474

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method Higueras, Inmaculada; Ketcheson, David I.; Kocsis, Tihamér A. Journal of Scientific Computing, Vol. 76, Issue 3 https://doi.org/10.1007/s10915-018-0664-3	journal	February 2018
Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal Journal of Scientific Computing, Vol. 81, Issue 3 https://doi.org/10.1007/s10915-019-01046-6	journal	September 2019
Evaluation of Implicit‐Explicit Additive Runge‐Kutta Integrators for the HOMME‐NH Dynamical Core Vogl, Christopher J.; Steyer, Andrew; Reynolds, Daniel R. Journal of Advances in Modeling Earth Systems, Vol. 11, Issue 12 https://doi.org/10.1029/2019ms001700	journal	December 2019
Evaluation of Implicit-Explicit Additive Runge-Kutta Integrators for the HOMME-NH Dynamical Core Vogl, Christopher J.; Steyer, Andrew; Reynolds, Daniel R. arXiv https://doi.org/10.48550/arxiv.1904.10115	text	January 2019
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97 MATHEMATICS AND COMPUTING
Implicit Runge–Kutta methods
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Strong stability preserving methods

Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

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