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

  • Christian Hendricks (a1), Matthias Ehrhardt (a1) and Michael Günther (a1)

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.

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*Corresponding author. Email addresses: (C. Hendricks) (M. Ehrhardt) (M. Günther)
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[14] C. C. W. Leentvaar and C. W. Oosterlee , Pricing multi-asset options with sparse grids and fourth order finite differences, Numerical Mathematics and Advanced Applications, Springer, (2006), pp. 975983.

[15] C. Reisinger , Analysis of linear difference schemes in the sparse grid combination technique, IMA J. Numer. Anal., 33(2) (2013), pp. 544581.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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