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

Journal Article · · Journal of Scientific Computing
 [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
+ Show Author Affiliations

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.

Research Organization:
Sandia National Lab. (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, Vol. 73, Issue 2-3; ISSN 0885-7474
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 20 works
Citation information provided by
Web of Science

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Cited By (4)

Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method journal February 2018
Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods journal September 2019
Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models journal January 2018
Evaluation of Implicit-Explicit Additive Runge-Kutta Integrators for the HOMME-NH Dynamical Core text January 2019

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97 MATHEMATICS AND COMPUTING
Ordinary differential equations
Partial differential equations
Strong stability preserving methods
Implicit Runge–Kutta methods
Implicit–explicit Runge–Kutta methods