Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

Ranocha, Hendrik; Sayyari, Mohammed; Dalcín, Lisandro; Parsani, Matteo; Ketcheson, David I.

doi:10.1137/19m1263480

Cited by 121 publications

(121 citation statements)

References 77 publications

(150 reference statements)

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“…The proposed methodology was presented in the context of continuous Galerkin methods. Further developments will focus on the DG version [1,2,8,27], extensions to high-order finite elements [3,22], and design of entropy stability preserving time integrators [28].…”

Section: Discussionmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable / entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

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Section: Discussionmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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“…• linearly covariant Furthermore, they do not require partitioning or temporal staggering, and they inherit other useful properties (such as strong stability preservation) of a selected Runge-Kutta method. Preservation of more general (non-inner-product) functionals is also possible with modification similar to that described herein; see [22]. The main contributions of the present work are: first, to further develop these methods that, while not entirely new, seem to have been overlooked; second, to put them on a rigorous footing in terms of accuracy and stability properties; and third, to explore their properties through analysis and numerical experiments.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

Relaxation Runge--Kutta Methods: Conservation and Stability for Inner-Product Norms

Ketcheson¹

2019

SIAM J. Numer. Anal.

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We further develop a simple modification of Runge-Kutta methods that guarantees conservation or stability with respect to any inner-product norm. The modified methods can be explicit and retain the accuracy and stability properties of the unmodified Runge-Kutta method. We study the properties of the modified methods and show their effectiveness through numerical examples, including application to entropy-stability for first-order hyperbolic PDEs. 1 We consider real spaces for simplicity, but the methods developed here are also applicable in complex spaces.

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Section: Introductionmentioning

confidence: 99%

“…It is possible to obtain conditional energy stability with explicit methods by using modifications that go outside the class of Runge-Kutta methods; e.g. projection methods [5,6,13] and relaxation Runge-Kutta schemes [18,29].Nevertheless, it is interesting to know what can be achieved within the class of explicit Runge-Kutta methods without modifications. In this setting, results have been obtained for problems that include a certain amount of dissipation [8,16].…”

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confidence: 99%

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

Ranocha

2020

IMA Journal of Numerical Analysis

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Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge-Kutta methods for nonlinear or non-autonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed and an example is given in which unconditional energy stability is obtained with an explicit scheme. IntroductionEver since the construction of numerical methods for ordinary and (time-dependent) partial differential equations (ODEs and PDEs, respectively), their stability has been an important and active topic of research. Monotonicity, meaning that the norm of the solution is bounded by its initial value, is a particularly exacting stability property. For equations, such as parabolic PDEs, that contain a significant amount of dissipation, any reasonable numerical method will typically preserve monotonicity under an appropriate time step restriction. In contrast, for non-dissipative problems such as hyperbolic PDEs (and their slightly dissipative semidiscretizations), common time discretizations may not preserve monotonicity under any finite step size.The energy method is an effective tool to get stability estimates, e.g. for hyperbolic PDEs [12,19]. Using summation by parts operators [9,34], these can be transferred efficiently to the semidiscrete level for many different kinds of schemes [11,20,21,28]. However, applying the same approach in time yields implicit methods [2,10,22,25]. Classical nonlinearly stable methods, such as algebraically stable Runge-Kutta methods, are also implicit. For hyperbolic problems, such implicit methods are usually less efficient than explicit ones. It is possible to obtain conditional energy stability with explicit methods by using modifications that go outside the class of Runge-Kutta methods; e.g. projection methods [5,6,13] and relaxation Runge-Kutta schemes [18,29].Nevertheless, it is interesting to know what can be achieved within the class of explicit Runge-Kutta methods without modifications. In this setting, results have been obtained for problems that include a certain amount of dissipation [8,16]. Recently this topic has again attracted the interest of researchers and several results (using the term strong stability) for linear, time-independent operators have been discovered [27,32,33]. Nonlinear problems have been investigated in [24], where many non-existence results for energy stable and strong stability 1 arXiv:1909.13215v1 [math.NA]

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Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

Cited by 121 publications

References 77 publications

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Relaxation Runge--Kutta Methods: Conservation and Stability for Inner-Product Norms

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

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