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Runge-Kutta methods and local uniform grid refinement

Authors: R. A. Trompert and J. G. Verwer
Journal: Math. Comp. 60 (1993), 591-616
MSC: Primary 65M50; Secondary 65L06, 65M20
MathSciNet review: 1181332
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Abstract: Local uniform grid refinement (LUGR) is an adaptive grid technique for computing solutions of partial differential equations possessing sharp spatial transitions. Using nested, finer-and-finer uniform subgrids, the LUGR technique refines the space grid locally around these transitions, so as to avoid discretization on a very fine grid covering the entire physical domain. This paper examines the LUGR technique for time-dependent problems when combined with static regridding. Static regridding means that in the course of the time evolution, the space grid is adapted at discrete times. The present paper considers the general class of Runge-Kutta methods for the numerical time integration. Following the method of lines approach, we develop a mathematical framework for the general Runge-Kutta LUGR method applied to multispace-dimensional problems. We hereby focus on parabolic problems, but a considerable part of the examination applies to hyperbolic problems as well. Much attention is paid to the local error analysis. The central issue here is a "refinement condition" which is to underly the refinement strategy. By obeying this condition, spatial interpolation errors are controlled in a manner that the spatial accuracy obtained is comparable to the spatial accuracy on the finest grid if this grid would be used without any adaptation. A diagonally implicit Runge-Kutta method is discussed for illustration purposes, both theoretically and numerically.

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  • [1] S. Adjerid and J. E. Flaherty, A local refinement finite element method for two dimensional parabolic systems, SIAM J. Sci. Statist. Comput. 9 (1988), 792-811. MR 957472 (89g:65136)
  • [2] D. C. Arney and J. E. Flaherty, An adaptive local mesh refinement method for time-dependent partial differential equations, Appl. Numer. Math. 5 (1989), 257-274. MR 1005377 (90h:65196)
  • [3] M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53 (1984), 484-512. MR 739112 (85h:65211)
  • [4] M. Crouzeix and P. A. Raviart, Approximation des problèmes d'évolution. 1: Etude des méthodes linéaires a pas multiples et des méthodes de Runge-Kutta, unpublished lecture notes, Université de Rennes, France.
  • [5] K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland, Amsterdam-New York-Oxford, 1984. MR 774402 (86g:65003)
  • [6] W. D. Gropp, Local uniform mesh refinement on vector and parallel processors, Large Scale Scientific Computing (P. Deuflhard and B. Engquist, eds.), Progr. Sci. Comput., Birkhäuser, Boston, 1987, pp. 349-367. MR 904514 (88m:65186)
  • [7] -, Local uniform mesh refinement with moving grids, SIAM J. Sci. Statist. Comput. 8 (1987), 292-304. MR 883772 (88f:65161)
  • [8] J. M. Sanz-Serna and J. G. Verwer, Stability and convergence at the PDE/stiff ODE interface, Appl. Numer. Math. 5 (1989), 117-132. MR 979551 (90c:65126)
  • [9] J. M. Sanz-Serna, J. G. Verwer, and W. H. Hundsdorfer, Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1987), 405-418. MR 875165 (88f:65146)
  • [10] R. A. Trompert and J. G. Verwer, Runge-Kutta methods and local uniform grid refinement, Report NM-R9022, Centre for Mathematics and Computer Science, Amsterdam, preprint. MR 1181332 (93h:65119)
  • [11] -, A static-regridding method for two dimensional parabolic partial differential equations, Appl. Numer. Math. 8 (1991), 65-90. MR 1128618 (92h:65146)
  • [12] -, Analysis of the implicit Euler local uniform grid refinement method, SIAM J. Sci. Statist. Comput. 14 (1993). MR 1204230 (93m:65139)

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Keywords: Partial differential equations, numerical mathematics, time-dependent problems, Runge-Kutta methods, adaptive grid methods, error analysis
Article copyright: © Copyright 1993 American Mathematical Society

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