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Open Access Research

On the solution of high order stable time integration methods

Owe Axelsson12, Radim Blaheta2, Stanislav Sysala2 and Bashir Ahmad1*

Author Affiliations

1 King Abdulaziz University, Jeddah, Saudi Arabia

2 IT4 Innovations Department, Institute of Geonics AS CR, Ostrava, Czech Republic

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Boundary Value Problems 2013, 2013:108  doi:10.1186/1687-2770-2013-108

Published: 26 April 2013

Abstract

Evolution equations arise in many important practical problems. They are frequently stiff, i.e. involves fast, mostly exponentially, decreasing and/or oscillating components. To handle such problems, one must use proper forms of implicit numerical time-integration methods. In this paper, we consider two methods of high order of accuracy, one for parabolic problems and the other for hyperbolic type of problems. For parabolic problems, it is shown how the solution rapidly approaches the stationary solution. It is also shown how the arising quadratic polynomial algebraic systems can be solved efficiently by iteration and use of a proper preconditioner.