On the solution of high order stable time integration methods
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RESEARCH
Open Access
On the solution of high order stable time integration methods Owe Axelsson1,2 , Radim Blaheta2 , Stanislav Sysala2 and Bashir Ahmad1* *
Correspondence: [email protected] 1 King Abdulaziz University, Jeddah, Saudi Arabia Full list of author information is available at the end of the article
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.
1 Introduction Evolution equations arise in many important practical problems, such as for parabolic and hyperbolic partial differential equations. After application of a semi-discrete Galerkin finite element or a finite difference approximation method, a system of ordinary differential equations, M
du + Au(t) = f(t), dt
t > , u() = u ,
arises. Here, u, f ∈ n , M is a mass matrix and M, A are n × n matrices. For a finite difference approximation, M = I, the identity matrix. In the above applications, the order n of the system can be very large. Under reasonable assumptions of the given source function f, the system is stable, i.e. its solution is bounded for all t > and converges to a fixed stationary solution as t → , independent of the initial value u . This holds if A is a normal matrix, that is, has a complete eigenvector space, and has eigenvalues with positive real parts. This condition holds for parabolic problems, where the eigenvalues of A are real and positive. In more involved problems, the matrix A may have complex eigenvalues with arbitrary large imaginary parts. Clearly, not all numerical time-integration methods preserve the above stability properties. Unless the time-step is sufficiently small, explicit time-integration methods do not converge and/or give unphysical oscillations in the numerical solution. Even with sufficiently small time-steps, algebraic errors may increase unboundedly due to the large number of time-steps. The simplest example where the stability holds is the Euler implicit method, ˜ + τ ) + τ Au(t ˜ + τ ) = u(t) ˜ + τ f(t + τ ), u(t
˜ t = τ , τ , . . . , u() = u˜ ,
© 2013 Axelsson et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Axelsson et al. Boundary Value Problems 2013, 2013:108 http://www.boundaryvalueproblems.com/content/2013/1/108
where τ > is th
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