A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Ranocha, Hendrik; Nordström, Jan

doi:10.1007/s10915-021-01454-7

Cited by 12 publications

(4 citation statements)

References 45 publications

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“…Also see [60,1,2] and references therein. Other applications of SBP operators include essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes [8,23,79], and implicit time integration methods [61,51,67].…”

Section: Introductionmentioning

confidence: 99%

Summation-by-parts operators for general function spaces

Glaubitz¹,

Nordström²,

Öffner³

2022

Preprint

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Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential and radial basis functions.

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Section: Introductionmentioning

confidence: 99%

Summation-by-parts operators for general function spaces

Glaubitz¹,

Nordström²,

Öffner³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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“…• nonlinear dispersive wave equations (Ranocha, Mitsotakis, et al, 2021;Ranocha, Quezada de Luna, et al, 2021) • hyperbolic conservation laws (LeFloch & Ranocha, 2021;Öffner & Ranocha, 2019; • ordinary differential equations Ranocha & Ketcheson, 2020;Ranocha & Nordström, 2021) • Helmholtz Hodge decomposition and analysis of plasma waves…”

Section: Related Research and Softwarementioning

confidence: 99%

SummationByPartsOperators.jl: A Julia library of provably stable discretization techniques with mimetic properties

Ranocha¹

2021

JOSS

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SummationByPartsOperators.jl is a Julia library of summation-by-parts (SBP) operators, which are discrete derivative operators developed to get provably stable (semi-) discretizations, paying special attention to boundary conditions. Discretizations included in this framework are finite difference, Fourier pseudospectral, continuous Galerkin, and discontinuous Galerkin methods. The main aim of SummationByPartsOperators.jl is to be useful for both students learning the basic concepts and researchers developing new numerical algorithms based on SBP operators. Therefore, SummationByPartsOperators.jl provides a unified framework of all of these seemingly different discretizations. At the same time, the implementation is reasonably optimized to achieve good performance without sacrificing flexibility.

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“…From a time discretization perspective, Nordström and Lundquist in [29] proposed SBP-based implicit time integrators to have fully discrete entropystable schemes. The work in [30,31] incorporated SBP in implicit Runge-Kutta methods. Friedrich et al [32] proposed entropy-stable space-time methods.…”

Section: Introductionmentioning

confidence: 99%

Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods

Kang¹,

Constantinescu²

2021

Preprint

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We propose entropy-preserving and entropy-stable partitioned Runge-Kutta (RK) methods. In particular we develop entropy conditions for implicit-explicit methods and a class of second-order multirate methods. We extend relaxation ideas for explicit methods to partitioned RK methods. We show that the proposed methods support fully entropy-preserving and entropy-stability properties at a discrete level. Numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the behavior of these methods.

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A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Cited by 12 publications

References 45 publications

Summation-by-parts operators for general function spaces

Summation-by-parts operators for general function spaces

SummationByPartsOperators.jl: A Julia library of provably stable discretization techniques with mimetic properties

Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods

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