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Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

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Abstract

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index-1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK

    Thorsten Sickenberger

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8-10, 1040, Wien, Austria

    Ewa Weinmüller

  3. Bergische Universität Wuppertal, Fachbereich C-Mathematik, Gaußstraße 20, 42199, Wuppertal, Germany

    Renate Winkler

Authors
  1. Thorsten Sickenberger
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  2. Ewa Weinmüller
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  3. Renate Winkler
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Corresponding author

Correspondence to Ewa Weinmüller.

Additional information

Communicated by Gustaf Söderlind.

The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.

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Sickenberger, T., Weinmüller, E. & Winkler, R. Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise. Bit Numer Math 49, 217–245 (2009). https://doi.org/10.1007/s10543-009-0209-0

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  • DOI: https://doi.org/10.1007/s10543-009-0209-0

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