Implicit and implicit-explicit strong stability preserving Runge–Kutta methods with high linear order

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

doi:10.1063/1.4992752

Cited by 6 publications

(24 citation statements)

References 26 publications

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“…If we consider the effective observed time-step (i.e. the observed time-step normalized by the number of stages s) we see that the SSPIFRK (9,4) has C ef f obs = 0.47 while the SSPIF-TSRK (3,4) has a smaller C ef f obs = 0.42 and SSPIF-TSRK(4, 4) has an even smaller C ef f obs = 0.4. However, we observe that the allowable TVD time-step of the SSPIFRK methods with (s, p) = (5, 4), (9,4) is smaller than that of the corresponding SSPIF-TSRK methods.…”

Section: Examplementioning

confidence: 99%

“…where∆ t FE << ∆t FE . For such cases, where nonlinear non-innerproduct stability properties are of concern, an implicit or implicit-explicit (IMEX) SSP scheme doesn't significantly alleviate the allowable time-step [4] and such methods will result in severe constraints on the allowable time-step [25,19,6,9,4].…”

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

“…In this section we consider, as we did in [15,16], a problem of the form (2.1) with a linear constant coefficient component Lu that satisfies the strong stability condition (2.3) and a nonlinear component N (u) that satisfies the strong stability condition (2.2). When∆t FE << ∆t FE using an explicit SSP Runge-Kutta method will result in severe constraints on the allowable time-step that is not alleviated by using an implicit or an implicit-explicit (IMEX) SSP Runge-Kutta method [25,19,6,9,4]. This is in contrast to the case with linear L 2 stability, where using an implicit or an IMEX method may completely alleviate the time-step restriction coming from the stiff component.…”

Section: A Review Of Explicit Ssp Two-stepmentioning

confidence: 99%

“…We consider a selection of fourth order methods. First, we explore the performance of the SSPIF-TSRK(s, p) methods with (s, p) = (3,4), (4,4) (the corresponding SSPIFRK methods do not exist). If we consider the effective observed time-step (i.e.…”

Section: Examplementioning

confidence: 99%

“…Unlike for smooth problems, where implicit methods (or implicit treatment of stiff terms) can eliminate the timestep restriction needed for linear or nonlinear inner-product norm stability, a strong stability preserving general linear method (GLM) of order greater than one has a finite SSP time-step [32]. In fact, it has been shown [25,6,19,4] that the SSP timestep restrictions for implicit and implicit-explicit (IMEX) SSP Runge-Kutta methods have a restrictive observed bound of C ≤ 2s . The limitation on the SSP coefficient of implicit and IMEX SSP Runge-Kutta methods led to the study of SSP integrating factor Runge-Kutta methods in [15].…”

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confidence: 99%

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Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Isherwood

Grant

2019

J Sci Comput

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Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge-Kutta methods were established in [15], where it was shown that it is possible to define explicit integrating factor Runge-Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge-Kutta method that has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods have an order barrier of p = 4, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge-Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge-Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP two-step Runge-Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge-Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge-Kutta methods designed in [15]. This paper is dedicated to the memory of Saul Abarbanel. His wisdom, humor, and kindness were a gift to all who knew him.

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Section: Examplementioning

confidence: 99%

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

Section: A Review Of Explicit Ssp Two-stepmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

mentioning

confidence: 99%

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Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Isherwood

Grant

2019

J Sci Comput

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TVD-MOOD schemes based on implicit-explicit time integration

Michel-Dansac

Thomann

2022

Applied Mathematics and Computation

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Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Isherwood¹,

Grant²

2018

SIAM J. Numer. Anal.

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Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of integrating factor Runge-Kutta methods have not been established. In this work we show that it is possible to define explicit integrating factor Runge-Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for an explicit integrating factor Runge-Kutta method to be SSP, namely that they are based on explicit SSP Runge-Kutta methods with non-decreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test-cases our new explicit SSP integrating factor Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta methods, implicit-explicit SSP Runge-Kutta methods, and some well-known exponential time differencing methods.

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Implicit and implicit-explicit strong stability preserving Runge–Kutta methods with high linear order

Cited by 6 publications

References 26 publications

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

TVD-MOOD schemes based on implicit-explicit time integration

Strong Stability Preserving Integrating Factor Runge--Kutta Methods

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