On the solution of high order stable time integration methods
1 King Abdulaziz University, Jeddah, Saudi Arabia
2 IT4 Innovations Department, Institute of Geonics AS CR, Ostrava, Czech Republic
Boundary Value Problems 2013, 2013:108 doi:10.1186/1687-2770-2013-108Published: 26 April 2013
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.