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

Abstract

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 plin ≤ 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 plin ≤ 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 behaviormore » of these methods on test problems.« less

Authors:
 [1];  [2];  [2];  [3]
  1. Univ. of Massachusetts, North Dartmouth, MA (United States). Dept. of Mathematics; Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Computational Mathematics Dept.
  2. Univ. of Massachusetts, North Dartmouth, MA (United States). Dept. of Mathematics
  3. 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
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1574706
Report Number(s):
SAND-2019-8924J
Journal ID: ISSN 0885-7474; 678045
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Scientific Computing
Additional Journal Information:
Journal Volume: 73; Journal Issue: 2-3; Journal ID: ISSN 0885-7474
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Ordinary differential equations; Partial differential equations; Strong stability preserving methods; Implicit Runge–Kutta methods; Implicit–explicit Runge–Kutta methods

Citation Formats

Conde, Sidafa, Gottlieb, Sigal, Grant, Zachary J., and Shadid, John N. Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order. United States: N. p., 2017. Web. doi:10.1007/s10915-017-0560-2.
Conde, Sidafa, Gottlieb, Sigal, Grant, Zachary J., & Shadid, John N. Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order. United States. doi:10.1007/s10915-017-0560-2.
Conde, Sidafa, Gottlieb, Sigal, Grant, Zachary J., and Shadid, John N. Mon . "Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order". United States. doi:10.1007/s10915-017-0560-2. https://www.osti.gov/servlets/purl/1574706.
@article{osti_1574706,
title = {Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order},
author = {Conde, Sidafa and Gottlieb, Sigal and Grant, Zachary J. and Shadid, John N.},
abstractNote = {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 plin ≤ 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 plin ≤ 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.},
doi = {10.1007/s10915-017-0560-2},
journal = {Journal of Scientific Computing},
number = 2-3,
volume = 73,
place = {United States},
year = {2017},
month = {9}
}

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