{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 2 0 1 3320a where ms s y i1 ci 3320b di

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ly, the quadrature rule can be designed to integrate polynomials exactly to as high a degree as possible. The actual ^ series expansion is complicated by the fact that F (s) is evaluated at ( ) in (3.3.21b). Isaacson and Keller 19] provide additional details on this matter however, we'll sidestep ^ the subtleties by assuming that all derivatives of ( ) are bounded so that F (s) has an expansion in powers of of the form ^ F (s)( ) = 0 + 1 +:::+ 33 r 1 ; r ; 1 + O( r ): d Pd(x) 01 1x 2 x2 ; 1 3 3 x3 ; 35x 3 4 x4 ; 6x2 + 35 7 5 x5 ; 109x3 + 5x 21 Table 3.3.2: Legendre polynomials Pd(x) of degree d 2 0 5] on ;1 x 1. The rst r terms of this series will be annihilated by (3.3.21b) if Ms ( ) is orthogonal to polynomials of degree r ; 1, i.e., if Z1 M ( ) q 1d = 0 q = 1 2 : : : r: (3.3.22) ; 0 Under these conditions, were we to transform the integrals in (3.3.21) and (3.3.22) back to t dependence using (3.3.13c), we would obtain the error of (3.3.18b) with p = s + r. With the s coe cients ci, i = 1 2 : : : s, we would expect the maximum value of r to be s. According to Theorem 3.3.6, this choice would lead to a collocation method of order 2s, i.e., a method having p = r + s = 2s and an O(h2s+1) local error. These are Butcher's maximal order formulas (Theorem 3.3.1) corresponding to the diagonal Pade approxi...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online