Stable Difference Methods for Block-Oriented Adaptive Grids
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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.
KeywordsSummation-by-parts Simultaneous-approximating-term Block-structured grid Adaptive mesh refinement Time-dependent Schrödinger equation Advection equation
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.
- 1.Appelö, D., Petersson, N.A.: A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun. Comput. Phys. 5, 84–107 (2008)Google Scholar
- 4.Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111, 220–236 (1994)Google Scholar
- 9.Kormann, K.: A time-space adaptive method for the Schrödinger equation. Tech. Rep. 2012–023. Department of Information Technology, Uppsala University (2012)Google Scholar
- 14.Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)Google Scholar
- 26.Tannor, D.J.: Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Book, Mill Valley (2007)Google Scholar