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Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique

Part of: Numerical approximation and computational geometry Partial differential equations, boundary value problems

Published online by Cambridge University Press:  20 June 2017

Christian Hendricks ,
Matthias Ehrhardt  and
Michael Günther
Christian Hendricks*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
Matthias Ehrhardt*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
Michael Günther*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
*
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
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Abstract

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Keywords

Combination techniquesparse gridsmultivariate interpolationfinite difference schemes

MSC classification

Secondary: 65N06: Finite difference methods 65D05: Interpolation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 689 - 710
Copyright
Copyright © Global-Science Press 2017 

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References

[1] De Boor, C., A Practical Guide to Splines, Springer, (1978).Google Scholar
[2] Bramble, J. H. and Hubbard, B. E., New monotone type approximations for elliptic problems, Math. Comp., 18 (1964), pp. 349–367.Google Scholar
[3] Bungartz, H. J. and Griebel, M., Sparse grids, Cambridge University Press, (2004), pp. 1–123.Google Scholar
[4] Bungartz, H. J. and Griebel, M. and Röschke, D. and Zenger, C., Pointwise convergence of the combination technique for Laplace's equation, East-West J. Numer. Math., 2 (1994), pp. 21–45.Google Scholar
[5] Bungartz, H. J. and Griebel, M. and Rüde, U., Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems, Comput. Methods Appl. Mech. Engrg., 116 (1994), pp. 243–252.Google Scholar
[6] Ciarlet, P. G., Discrete maximum principle for finite-difference operators, Aequationes Mathematicae, 4(3) (1970), pp. 338–352.Google Scholar
[7] Gaikwad, A. and Toke, I. M., GPU based sparse grid technique for solving multidimensional options pricing PDEs, The Workshop on High Performance Computational Finance, (2009), pp. 1–9.Google Scholar
[8] Garcke, J., Sparse Grid Tutorial, Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Mathematical Sciences Institute, (2006).Google Scholar
[9] Griebel, M. and Hamaekers, J., MSparse grids for the Schrödinger equation, ESAIM-Math. Model. Num., 41(2) (2007), pp. 215–247.CrossRefGoogle Scholar
[10] Griebel, M. and Schneider, M. and Zenger, C., A combination technique for the solution of sparse grid problems, IMACS Elsevier, Iterative Methods in Linear Algebra, (1992), pp. 263–281.Google Scholar
[11] Griebel, M. and Thurner, V., The efficient solution of fluid dynamics problems by the combination technique, Int. J. Numer. Method H., 5(3) (1995), pp. 251–269.Google Scholar
[12] Hall, C. A., On Error Bounds for Spline Interpolation, J. Approx. Theory, 1968, pp. 209–218.Google Scholar
[13] Hendricks, C. and Ehrhardt, M., Evaluating the effects of changing market parameters and policy implications in the German electricity market, J. Energy Markets, 7(2) (2014).Google Scholar
[14] Leentvaar, C. C. W. and Oosterlee, C. W., Pricing multi-asset options with sparse grids and fourth order finite differences, Numerical Mathematics and Advanced Applications, Springer, (2006), pp. 975–983.CrossRefGoogle Scholar
[15] Reisinger, C., Analysis of linear difference schemes in the sparse grid combination technique, IMA J. Numer. Anal., 33(2) (2013), pp. 544–581.Google Scholar
[16] Reisinger, C., Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben, PhDthesis, Ruprecht-Karls-Universität, 2004.Google Scholar
[17] Zenger, C., Sparse grids, Parallel Algorithms for Partial Differential Equations, 31 (1991), pp. 241–251.Google Scholar