SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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On Numerical Properties of Accelerated Multiple Precision Fully Implicit Runge-Kutta Methods

T. Kouya

Abstract
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta (IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR [1] as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3 [2], of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method [3], (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods [4]. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as 1D Kuramoto-Sivashinsky equation.

Bibliography
[1] MPFR Project, The MPFR library. http://www.mpfr.org/
[2] L. O. Jay, SPARK3. http://www.math.uiowa.edu/ ljay/SPARK3.html
[3] A. Buttari, J. Dongarra, Julie Langou, Julien Langou, P. Luszczek, and J. Karzak, Mixed precision iterative refinement techniques for the solution of dense linear system, The International Journal of High Performance Computing Applications, Vol. 21, No. 4, pp. 457-466, 2007.
[4] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, Springer-Verlag, 1996.

Organized by         Universidad de Valladolid     IMUVA