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

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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.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Louisiana State University, 202 Nicholson Hall, Baton Rouge, LA, 70803, USA

    Peter Diener, Ernst Nils Dorband & Manuel Tiglio

  2. Center for Computation and Technology, Louisiana State University, 302 Johnston Hall, Baton Rouge, LA, 70803, USA

    Peter Diener, Ernst Nils Dorband, Erik Schnetter & Manuel Tiglio

  3. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476, Golm, Germany

    Erik Schnetter

Authors
  1. Peter Diener
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  2. Ernst Nils Dorband
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  3. Erik Schnetter
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  4. Manuel Tiglio
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Correspondence to Manuel Tiglio.

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Diener, P., Dorband, E.N., Schnetter, E. et al. Optimized High-Order Derivative and Dissipation Operators Satisfying Summation by Parts, and Applications in Three-dimensional Multi-block Evolutions. J Sci Comput 32, 109–145 (2007). https://doi.org/10.1007/s10915-006-9123-7

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