Summation-by-parts operators for correction procedure via reconstruction

Ranocha, Hendrik; Öffner, Philipp; Sonar, Thomas

doi:10.1016/j.jcp.2016.02.009

Cited by 128 publications

(171 citation statements)

References 28 publications

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“…where the SBP property (2) has been used in the last equality. Hence, inserting (32) The second factor is non-negative because F is positive semidefinite with respect to the scalar product induced by M, cf. Lemma 3.9.…”

Section: Linear Stabilitymentioning

confidence: 99%

“…Based on the fact that integration by parts plays a major role in the development of energy and entropy estimates for initial boundary value problems, one may conjecture that the summation by parts (SBP) property [7,43] is a key factor in provably stable schemes. Although it is complicated to formulate such a conjecture mathematically, there are several attempts to unify stable methods in the framework of summation by parts schemes, starting from the origin of SBP operators in finite difference methods [14,38] and ranging from finite volume [19,20] and discontinuous Galerkin methods [8] to flux reconstruction schemes [32].…”

Section: Introductionmentioning

confidence: 99%

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Some notes on summation by parts time integration methods

Ranocha

2019

Results in Applied Mathematics

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Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is necessarily involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods including the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

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Section: Linear Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some notes on summation by parts time integration methods

Ranocha

2019

Results in Applied Mathematics

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“…For all other viable choices of i and j, multiplying (20) from the left by (e i ) T , then adding the result with i and j swapped yields (21) (i + j)1 TP e i+j−1 = (e i ) T Sf j + (e j ) T Sf i .…”

Section: Appendix C Proof Of Lemmamentioning

confidence: 99%

“…These include spectral collocation and spectral element methods [2,8,9] as well as correction procedures via reconstruction [21]. Further, the finite difference class of SBP operators has been enlarged to multidimensional operators similar to Galerkin methods [12] as well as to grid dependent stencils akin to element based methods [5].…”

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confidence: 99%

On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form

Linders¹,

Lundquist²,

Nordström³

2018

SIAM J. Numer. Anal.

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Abstract. In this paper we generalise results regarding the order of accuracy of finite difference operators on Summation-By-Parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.Key words. Finite difference schemes, summation-by-parts operators, numerical differentiation, quadrature rules, order of accuracy AMS subject classifications. 65M06, 65N061. Introduction. Summation-By-Parts (SBP) operators, applied in the discretisation of systems of partial differential equations have received considerable attention since they lead to provable energy stability [25], and recently, entropy stability [6] for well-posed problems. Finite difference stencils on SBP form were first introduced in [14,15] based on central difference schemes of order 2 and 4. Later, operators with minimal bandwidth using stencils of order 6 and 8 were developed in [22]. In [3,19] SBP operators of orders up to 8 for both first and second derivatives were presented.The SBP concept has been extended to methods outside the finite difference community. These include spectral collocation and spectral element methods [2,8,9] as well as correction procedures via reconstruction [21]. Further, the finite difference class of SBP operators has been enlarged to multidimensional operators similar to Galerkin methods [12] as well as to grid dependent stencils akin to element based methods [5]. In this paper we restrict our attention to fixed stencil finite difference schemes and do not consider these extended approaches further.Implicit to the definition of an SBP operator is the notion of a discrete norm. If this norm is represented by a diagonal matrix, the associated operator is referred to as a diagonal norm based SBP operator. To avoid stability issues on curvilinear grids [23], and in general for problems with variable coefficients [20], finite difference operators on SBP form are in practice usually based on a diagonal norm.The focus in this paper is on SBP operators consisting of a repeated central difference stencil in the interior, and one-sided stencils near boundaries and interfaces. We introduce the notation SBP(τ ,2s) to refer to such an operator that is of order τ near the boundary and of order 2s in the interior. For completely uniform grid distributions, the accuracy of SBP(τ ,2s) is known to be dictated by two main constraints; we will formalise these as Theorems 4 and 5 in the next section. The first one states that s ≥ τ , i.e. the order of accuracy of the int...

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“…By mimicking certain properties of the underlying PDEs, an SBP-SAT discretisation yields a numerical solution that satisfies bounds analogous to the true solution of the well-posed governing equations. Although originally proposed for finite difference methods [9], the SBP framework has been extended to methods outside this paradigm, including finite volume methods [12,13], spectral Galerkin and element methods [3,6], correction procedures via reconstruction [15] and temporal discretisations [14].…”

Section: Introductionmentioning

confidence: 99%

Error analysis of summation-by-parts formulations : Dispersion, transmission and accuracy

Linders¹

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In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximationterms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems.The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times.In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid.In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations. D R A F T D R A F T Sammanfattning på svenskaPartiella differentialekvationer ligger till grund för matematiska modeller av mångaämnesspecifika problem inom t.ex. fysik, kemi, ekonomi och meteorologi. Vanligen angrips sådana problem numeriskt genom att ersätta de styrande ekvationerna med en lämplig diskret approximation, vilken i sin tur kan lösas med hjälp av en dator. Detär emellertid oundvikligt att den numeriska proceduren introducerar fel i lösningen. I allmänhetär det utmanande att analysera dessa fel, samt att både begränsa och minimera dem. I denna avhandling beaktas just sådana utmaningar.En method som begränsar tillväxten av numeriska fel sägs vara stabil. För finita differensapproximationer på partiell summationsform (eng. "summationby-parts", SBP) kan stabilitet systematiskt visas om rand-och begynnelsevillkor implementeras på svag form. Att minimera felenär emellertid ett separat problem och beror på de styrande ekvationernas natur.Avhandlin...

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Summation-by-parts operators for correction procedure via reconstruction

Cited by 128 publications

References 28 publications

Some notes on summation by parts time integration methods

Some notes on summation by parts time integration methods

On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form

Error analysis of summation-by-parts formulations : Dispersion, transmission and accuracy

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