Downwinding for preserving strong stability in explicit integrating factor Runge–Kutta methods

Isherwood, Leah; Grant, Zachary J.

doi:10.4310/pamq.2018.v14.n1.a1

Cited by 4 publications

(9 citation statements)

References 15 publications

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“…In [16] we show that it is not absolutely necessary to have non-decreasing abscissas: even if there are some decreasing abscissas one can preserve the strong stability property by replacing the operator L by an operator L that satisfies…”

Section: Ssp Analysis Of Integrating Factormentioning

confidence: 96%

“…if L is a constant coefficient operator), the cost may be reasonable, but in some cases the exponential must be computed at every step, and low-storage, matrix-free approaches are needed. New approaches to efficiently compute the matrix exponential have been recently considered in [1,31,27,7], and such approaches, as well as others, will be critical for bringing the integrating factor methods proposed in [15,16] and the current work into practical use.…”

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

“…It is important to note that in the current work as well as our prior work [15,16], the cost of computation of the matrix exponential will be a critical factor in the ability to efficiently implement the proposed methods. If the computation of the matrix exponential is only needed once per simulation (i.e.…”

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

“…Explicit SSP two-step Runge-Kutta schemes for use with integrating factor methods. 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].…”

Section: 2mentioning

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: Ssp Analysis Of Integrating Factormentioning

confidence: 96%

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

Section: Efficient Computation Of the Matrix Exponentialmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 2 more Smart Citations

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

Isherwood

Grant

2019

J Sci Comput

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“…Hence, direct applications of these schemes to hyperbolic PDEs with discontinuous solutions may lead to nonlinear instability. Recently, integrating factor Runge-Kutta (IFRK) methods with a SSP property were developed in [38][39][40] to solve one-dimensional hyperbolic PDEs, which have explicit linear and nonlinear parts. These methods are exponential integrators which satisfy the SSP requirement.…”

Section: Introductionmentioning

confidence: 99%

Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Chen

2021

Mathematics

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Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Efficient inequality-preserving integrators for differential equations satisfying forward Euler conditions

Zhang¹,

Xu²,

Xia³

et al. 2023

ESAIM: M2AN

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Developing explicit, high-order accurate, and stable algorithms for nonlinear differential equations remains an exceedingly difficult task. In this work, a systematic approach is proposed to develop high-order, large time-stepping schemes that can preserve inequality structures shared by a class of differential equations satisfying forward Euler conditions. Strong-stability-preserving (SSP) methods are popular and effective for solving equations of this type. However, few methods can deal with the situation when the time-step size is larger than that allowed by SSP methods. By adopting time-step-dependent stabilization and taking advantage of integrating factor methods in the Shu-Osher form, we propose enforcing the inequality structure preservation by approximating the exponential function using a novel recurrent approximation without harming the convergence. We define sufficient conditions for the obtained parametric Runge-Kutta (pRK) schemes to preserve inequality structures for any time-step size, namely, the underlying Shu-Osher coefficients are non-negative. To remove the requirement of a large stabilization term caused by stiff linear operators, we further develop inequality-preserving parametric integrating factor Runge-Kutta (pIFRK) schemes by incorporating the pRK with an integrating factor related to the stiff term, and enforcing the non-decreasing of abscissas. The only free parameter can be determined a priori based on the SSP coefficient, the time-step size, and the forward Euler condition. We demonstrate that the parametric methods developed here offer an effective and unified approach to study problems that satisfy forward Euler conditions, and cover a wide range of well-known models. Finally, numerical experiments reflect the high-order accuracy, efficiency, and inequality-preserving properties of the proposed schemes.

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Downwinding for preserving strong stability in explicit integrating factor Runge–Kutta methods

Cited by 4 publications

References 15 publications

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

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

Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Efficient inequality-preserving integrators for differential equations satisfying forward Euler conditions

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