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Abstract and Applied Analysis
Volume 2014, Article ID 140849, 15 pages
http://dx.doi.org/10.1155/2014/140849
Research Article

A Simplified Milstein Scheme for SPDEs with Multiplicative Noise

Department of Applied Mathematics, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran

Received 3 May 2014; Revised 21 June 2014; Accepted 29 June 2014; Published 4 August 2014

Academic Editor: Reza Ezzati

Copyright © 2014 B. Ghayebi and S. M. Hosseini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with a research question raised by Jentzen and Röckner (A Milstein scheme for SPDEs, arXiv:1001.2751v4 (2012)), whether the exponential term in their introduced scheme can be replaced by a simpler mollifier. This replacement can lead to more simplification and computational reduction in simulation. So, in this paper, we essentially replace the exponential term with a Padé approximation of order 1 and denote the resulting scheme by simplified Milstein scheme. The convergence analysis for this scheme is carried out and it is shown that even with this replacement the order of convergence is maintained, while the resulting scheme is easier to implement and slightly more efficient computationally. Some numerical tests are given that confirm the order of accuracy and also computational cost reduction.