Journal of Scientific Computing

, Volume 75, Issue 2, pp 906–940 | Cite as

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

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Abstract

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP–SAT). Recently it was shown that SBP–SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.

Keywords

Finite differences Summation by parts Simultaneous approximation term Dual consistency Superconvergence Narrow stencil 

Mathematics Subject Classification

65M06 65M20 65M12 

Notes

Acknowledgements

The author would like to sincerely thank the anonymous referees for their valuable comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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