Abstract
In the numerical technique considered in this paper, time-stepping is performed on a set of semi-coarsened space grids. At given time levels the solutions on the different space grids are combined to obtain the asymptotic convergence of a single, fine uniform grid. We present error estimates for the two-dimensional, spatially constant-coefficient model problem and discuss numerical examples. A spatially variable-coefficient problem (Molenkamp-Crowley test) is used to assess the practical merits of the technique. The combination technique is shown to be more efficient than the single-grid approach, yet for the Molenkamp-Crowley test standard Richardson extrapolation is still more efficient than the combination technique. However, parallelization is expected to significantly improve the combination technique’s performance.
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References
C. Zenger, Sparse grids, in: W. Hackbusch, ed., Notes on Numerical Fluid Mechanics, Vol. 31, 241–251 (Vieweg, Braunschweig, 1991 ).
M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, in: R. Beauwens and P. de Groen, eds., Iterative Methods in Linear Algebra, 263–281 ( North-Holland, Amsterdam, 1992 ).
M. Griebel and G. Zumbusch, Adaptive sparse grids for hyperbolic conservation laws, in: M. Fey and R. Jeltsch, eds., Hyperbolic Problems: Theory, Numerics, Applications, International Series of Numerical Mathematics, Vol. 129 ( Birkhäuser, Basel, 1999 ).
B. Lastdrager, B. Koren and J. Verwer, The sparse-grid combination technique applied to time-dependent advection problems, Report (in print), CWI, Amsterdam 1999.
B. Lastdrager and B. Koren, Error analysis for function representation by the sparse-grid combination technique, Report MAS-R9823, CWI, Amsterdam 1998.
U. Rüde, Multilevel, extrapolation and sparse grid methods, in: P.W. Hemker and P. Wesseling, eds., Multigrid Methods IV, International Series of Numerical Mathematics, Vol. 116, 281–294 ( Birkhäuser, Basel, 1993 ).
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© 2000 Springer-Verlag Berlin Heidelberg
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Lastdrager, B., Koren, B., Verwer, J. (2000). The Sparse-Grid Combination Technique Applied to Time-Dependent Advection Problems. In: Dick, E., Riemslagh, K., Vierendeels, J. (eds) Multigrid Methods VI. Lecture Notes in Computational Science and Engineering, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58312-4_19
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DOI: https://doi.org/10.1007/978-3-642-58312-4_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67157-2
Online ISBN: 978-3-642-58312-4
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