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Stable Difference Methods for Block-Oriented Adaptive Grids

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Abstract

In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP–SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP–SAT framework, with a reduced number of SBP–SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.

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Acknowledgments

The authors would like to thank Sverker Holmgren and Gunilla Kreiss for valuable insight and discussions. The design of the interpolation operators is based on a Maple sheet by Ken Mattsson. The simulations were performed on resources provided by SNIC-UPPMAX under Projects p2003013 and p2005005.

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Anna Nissen

  2. Zentrum Mathematik, Technische Universität München, Garching, Germany

    Katharina Kormann

  3. Department of Information Technology, Uppsala University, Uppsala, Sweden

    Magnus Grandin & Kristoffer Virta

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  1. Anna Nissen
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  2. Katharina Kormann
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  3. Magnus Grandin
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  4. Kristoffer Virta
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Corresponding author

Correspondence to Anna Nissen.

Appendix: Interpolation Operators at SBP-FD Junctions

Appendix: Interpolation Operators at SBP-FD Junctions

The part of \(\left( \begin{array}{c c} I_w^{u} \\ I_w^{v} \\ \end{array}\right) \) around the interface is given by for order 4

$$\begin{aligned} \left( \begin{array}{c c} \bar{I}_w^{u} \\ \bar{I}_w^{v} \\ \end{array}\right) =\left( \begin{array}{c c c c c c c c} 1&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0\\ 0&{} 1&{} 0&{} 0&{} 0&{} 0&{} 0\\ -\frac{3}{59}&{} \frac{10}{59}&{} \frac{48}{59}&{} \frac{4}{59}&{} 0 &{}0&{} 0\\ \frac{2}{17} &{}-\frac{5}{17}&{} \frac{2}{17}&{} \frac{20}{17}&{} -\frac{2}{17}&{} 0&{} 0\\ 0&{} 0&{} -\frac{2}{17} &{}\frac{20}{17} &{}\frac{2}{17}&{} -\frac{5}{17}&{} \frac{2}{17}\\ 0&{} 0 &{}0 &{}\frac{4}{59}&{} \frac{48}{59}&{} \frac{10}{59}&{} -\frac{3}{59}\\ 0&{} 0&{} 0&{} 0&{} 0&{} 1&{} 0\\ 0 &{}0&{} 0&{} 0&{} 0&{} 0&{} 1\\ \end{array}\right) , \end{aligned}$$
(42)

and by for order 6,

$$\begin{aligned} \left( \begin{array}{c c} \bar{I}_w^{u} \\ \bar{I}_w^{v} \\ \end{array}\right)&=\left( \begin{array}{c c c c c c c c c c c c c} 1&{} 0&{} 0&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 0 &{}0 &{}0\\ 0&{} 1&{} 0&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 0 &{}0 &{}0\\ 0&{} 0&{} 1&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 0 &{}0 &{}0\\ \frac{-601}{2711}&{} \frac{2289}{2711}&{} -\frac{3117}{2711} &{} \frac{4320}{2711} &{} 0 &{} -\frac{198}{2711}&{} \frac{18}{2711}&{} 0&{} 0&{} 0&{} 0\\ \frac{1803}{12013} &{} -\frac{6104}{12013}&{} \frac{6234}{12013}&{} 0&{} \frac{8604}{12013} &{} \frac{1584}{12013} &{} -\frac{108}{12013}&{} 0&{} 0&{} 0&{} 0 \\ \frac{-3606}{13649}&{} \frac{11445}{13649}&{} -\frac{10390}{13649}&{} 0&{} \frac{1980}{13649} &{} \frac{15660}{13649} &{} -\frac{1440}{13649}&{} 0&{} 0&{} 0&{} 0\\ 0&{} 0&{} 0&{} 0 &{} -\frac{1440}{13649} &{} \frac{15660}{13649}&{} \frac{1980}{13649} &{}-\frac{10390}{13649}&{} 0&{} \frac{11445}{13649}&{} -\frac{3606}{13649} \\ 0&{} 0&{} 0&{} 0 &{} -\frac{108}{12013} &{} \frac{1584}{12013}&{} \frac{8604}{12013}&{} \frac{6234}{12013}&{} 0&{} -\frac{6104}{12013}&{} \frac{1803}{12013} \\ 0&{} 0&{} 0&{} 0 &{} \frac{18}{2711} &{} -\frac{198}{2711}&{} 0 &{} \frac{4320}{2711}&{}-\frac{3117}{2711}&{} \frac{2289}{2711} &{} -\frac{601}{2711} \\ 0&{} 0&{} 0&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 0 &{}1 &{}0\\ 0&{} 0&{} 0&{} 0&{} 0 &{} 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array}\right) . \end{aligned}$$

\(I_{uv}^w\) is then given by

$$\begin{aligned} I_{uv}^w = \left( \begin{array}{cc} P_{x,w}^{-1}(I_w^u)^T P_{x,u}&P_{x,w}^{-1}(I_w^v)^T P_{x,v} \end{array} \right) . \end{aligned}$$

For interpolation operators for order 8, please contact the authors.

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Nissen, A., Kormann, K., Grandin, M. et al. Stable Difference Methods for Block-Oriented Adaptive Grids. J Sci Comput 65, 486–511 (2015). https://doi.org/10.1007/s10915-014-9969-z

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