Optimal diagonal-norm SBP operators

Mattsson, Ken; Almquist, Martin; Carpenter, Mark H.

doi:10.1016/j.jcp.2013.12.041

Cited by 43 publications

(61 citation statements)

References 51 publications

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“…This is in line with the convergence results observed in [17]. This result is thus a generalisation of the classical Theorem 4 to the arbitrary non-uniform grid (1).…”

Section: Generalisation Of Theoremsupporting

confidence: 91%

“…However, both results assume the use of uniform grids. With the emergence of diagonal norm based SBP operators defined in non-uniform settings [17,4], generalisations of Theorems 4 and 5 are needed. This is the goal of the next three sections.…”

Section: 3mentioning

confidence: 99%

“…However, in recent years finite difference operators on SBP form defined on grids with non-uniform point distributions near boundaries and interfaces have emerged. Significant error reductions have been observed for several model problems using such operators [17]. Further, operators defined on grids that do not match with the physical domain boundaries have been introduced in [4].…”

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confidence: 99%

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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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“…This is in line with the convergence results observed in [17]. This result is thus a generalisation of the classical Theorem 4 to the arbitrary non-uniform grid (1).…”

Section: Generalisation Of Theoremsupporting

confidence: 91%

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…Remark Starting from (21) and assuming periodic boundary conditions (here referred to as the Cauchy problem), we can simultaneously diagonalise H −1 C and H −1 A, and for each Fourier mode derive a scalar ODE of the form given by (22). The Cauchy problem do share much of the characteristics with the corresponding IBVP, when it comes to expected the CFL condition.…”

Section: Explicit Runge-kutta Methodsmentioning

confidence: 99%

High-fidelity numerical simulation of the dynamic beam equation

Mattsson

Stiernström

2015

Journal of Computational Physics

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PostprintThis is the accepted version of a paper published in Journal of Computational Physics. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.Citation for the original published paper (version of record):Mattsson, K., Stiernström, V. (2015) High-fidelity numerical simulation of the dynamic beam equation. Journal of Computational Physics

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“…The SBP-SAT method combines semi-discrete operators that satisfy a SBP formula [23], with phys-ical BC implemented using the simultaneous approximation term (SAT) method [9]. Recent examples of the SBP-SAT approach can be found in [20,28,2,38,13].…”

Section: Introductionmentioning

confidence: 99%

High-fidelity numerical simulation of solitons in the nerve axon

Mattsson

Werpers

2016

Journal of Computational Physics

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PostprintThis is the accepted version of a paper published in Journal of Computational Physics. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.Citation for the original published paper (version of record):Mattsson, K., Werpers, J. (2016) High-fidelity numerical simulation of solitons in the nerve axon. Journal of Computational Physics

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Optimal diagonal-norm SBP operators

Cited by 43 publications

References 51 publications

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

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

High-fidelity numerical simulation of the dynamic beam equation

High-fidelity numerical simulation of solitons in the nerve axon

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