39d for an s stage runge kutta method applied to an m

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: direct product of two matrices as 2 3 a11 J a12J a1sJ 6 a21 J a22 J a2sJ 7 6 7 A J = 6 .. (3.3.10b) ... . . . ... 7 : 4. 5 as1J as2J assJ Then, (3.3.9b) can be written concisely as (I ; hA J) Y( ) = ;F( ) where A was given by (3.2.6c) and 2 () Y1 ( 6 Y2 ) 6 () Y = 6 .. 4. Ys( ) 3 7 7 7 5 2 6 () F =6 6 4 (3.3.10c) 3 F(1 ) F(2 ) 7 : 7 ... 7 5 () Fs (3.3.10d) The approximation of the Jacobian does not change the accuracy of the computed solution, only the convergence rate of the iteration. As long as convergence remains good, the same Jacobian can be used for several time step and only be re-evaluated when convergence of the Newton iteration slows. Even with this simpli cation, with m ranging into the thousands, the solution of (3.3.10) is clearly expensive and other ways of reducing the computational cost are necessary. Diagonally implicit Runge-Kutta (DIRK) methods o er one possibility. A DIRK method is one where aij = 0, i < j and at least one aii 6= 0, i j = 1 2 : : : s. If, in addition, a11 = a22 = : : : = ass = a, the techniq...
View Full Document

This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online