Some notes on summation by parts time integration methods

Ranocha, Hendrik

doi:10.1016/j.rinam.2019.100004

Cited by 19 publications

(17 citation statements)

References 43 publications

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“…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.…”

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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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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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“…However, transferring such semidiscrete results to fully discrete schemes is not easy in general. Stability/dissipation results for fully discrete schemes have mainly been limited to semidiscretizations including certain amounts of dissipation [31,32,49,71], linear equations [53,54,64,65,68], or fully implicit time integration schemes [7,10,11,26,36,39,48]. For explicit methods and general equations, there are negative experimental and theoretical results concerning energy/entropy stability [37,38,47,50].…”

Section: Related Workmentioning

confidence: 99%

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

Ranocha¹,

Sayyari²,

Dalcín³

et al. 2020

SIAM J. Sci. Comput.

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Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation We generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Navier-Stokes equations. Based on the unstructured hp-adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summation-by-parts and simultaneous-approximation-term operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily high-order accurate in space and time. We demonstrate the accuracy and the robustness of the fully-discrete explicit locally entropy-stable solver for a set of test cases of increasing complexity.

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“…A consistent derivative operator D satisfies span{1} ≤ ker D. Some undesired behaviour can occur if ker D span{1}, cf. [30,43,56].…”

Section: Remark 24mentioning

confidence: 99%

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

Ranocha

Ostaszewski

Heinisch

2020

Commun. Appl. Math. Comput.

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In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated to grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully.In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

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Some notes on summation by parts time integration methods

Cited by 19 publications

References 43 publications

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

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

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

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

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