Journal of Scientific Computing

, Volume 32, Issue 1, pp 109–145 | Cite as

Optimized High-Order Derivative and Dissipation Operators Satisfying Summation by Parts, and Applications in Three-dimensional Multi-block Evolutions

  • Peter Diener
  • Ernst Nils Dorband
  • Erik Schnetter
  • Manuel Tiglio
Article

We construct optimized high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators’ spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions. In particular, we find that the constructed dissipation operators are effective in suppressing instabilities that are sometimes otherwise present in the restricted full norm case.

Keywords

High order finite differencing numerical stability multi-block evolutions artificial dissipation accuracy 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Peter Diener
    • 1
    • 2
  • Ernst Nils Dorband
    • 1
    • 2
  • Erik Schnetter
    • 2
    • 3
  • Manuel Tiglio
    • 1
    • 2
  1. 1.Department of Physics and AstronomyLouisiana State UniversityBaton RougeUSA
  2. 2.Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  3. 3.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutGolmGermany

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