A solution to the stability issues with block norm summation by parts operators

Mattsson, Ken; Almquist, Martin

doi:10.1016/j.jcp.2013.07.013

Cited by 34 publications

(44 citation statements)

References 37 publications

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“…It is further shown [21] that 10th order diagonal-norm SBP operator require eleven boundary stencils to achieve optimal accuracy. In a previous paper [25] we derived a 10th order diagonal SBP operator using eleven boundary points. The accuracy is still very low compared to the internal scheme, and the operator is not attractive.…”

Section: The Finite Difference Methodsmentioning

confidence: 99%

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Optimal diagonal-norm SBP operators

Mattsson

Almquist

Carpenter

2014

Journal of Computational Physics

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Optimal boundary closures are derived for first derivative, finite difference operators of order 2, 4, 6 and 8. The closures are based on a diagonal-norm summation-by-parts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multi-block grids and entropy stability for nonlinear equations that support a convex extension. The new closures are developed by enriching conventional approaches with additional boundary closure stencils and non-equidistant grid distributions at the domain boundaries. Greatly improved accuracy is achieved near the boundaries, as compared with traditional diagonal norm operators of the same order. The superior accuracy of the new optimal diagonal-norm SBP operators is demonstrated for linear hyperbolic systems in one dimension and for the nonlinear compressible Euler equations in two dimensions.

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Section: The Finite Difference Methodsmentioning

confidence: 99%

“…Hence, we will solve the Euler-vortex problems on a twoblock domain (see Figures 7 and 7). Details concerning how to discretize this problem with the SBP-SAT method can be found in [44,31,25].…”

Section: The Euler Vortex Problemmentioning

confidence: 99%

Optimal diagonal-norm SBP operators

Mattsson

Almquist

Carpenter

2014

Journal of Computational Physics

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“…The truncation error of S is O(h p+1 ). Such operators are constructed for p = 1, 2, 3, 4 in [15], and p = 5 in [12].…”

Section: Remarkmentioning

confidence: 99%

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

Wang

Kreiss

2016

J Sci Comput

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When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re(s) ≥ 0, two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at s = 0. We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.

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“…, of the first and second derivatives in (21). The first derivative in the initial step can be approximated with…”

Section: First and Second Derivatives In Timementioning

confidence: 99%

“…By replacing the first derivative in the initial step by D + v 0 and inserting appropriate Taylor expansions we obtain the following second-order accurate approximation of (21), …”

Section: First and Second Derivatives In Timementioning

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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A solution to the stability issues with block norm summation by parts operators

Cited by 34 publications

References 37 publications

Optimal diagonal-norm SBP operators

Optimal diagonal-norm SBP operators

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

High-fidelity numerical simulation of the dynamic beam equation

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