Download Construction of Integration Formulas for Initial Value by P.J.Van Der Houwen PDF

By P.J.Van Der Houwen

ISBN-10: 0720423503

ISBN-13: 9780720423501

ISBN-10: 0720423724

ISBN-13: 9780720423723

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Example text

In a Taylor series about x = xn. 1. 2) Λχη + h») - h(y(Xn + hn), J>(*»)) J \ ' \x=xn + hn J υ \ / \x=xn = ( l - ^ o ) X ^ ) + (l+ai-/5i)/z„/(xw)+^(l+2ai + 2a2-2^2)^/Xxw) + 1(1+3αι + 6α2 + 6α 3 -6β 3 ) *«/"(*,,) + 2 ί(ΐ4-4αι+12α 2 + 24α3 + 2 4 α 4 - 2 4 ^ 4 ) ^ / ' , , ( ^ ) + 0 ( ^ ) . 1 is easily derived. This table should be interpreted such, that the conditions on ß09 ßl9 . . , βρ together, guarantee consistency of order p. The derivation of the conditions for fifth or higher order consistency are left to the reader.

1. 2, in connection with the application of Newton-Raphson iteration to the trapezoidal rule. 12"). 2) *"y> = h Θ■- A KJ J — χιιη, J ^ ^ j = 0, 1 , . . , m- 3, j? 3) 0j = 9j(h„J(y„)), AJt, = AJt i(h„J(y„)). Note that each integration step requires one evaluation of the Jacobian matrix 7. In particular, when the, coefficient functions A} t and Θ} are polynomials, then the method is said to be explicit; if at least one of these functions is not a polynomial the method is said to be semi-implicit.

10') in a different form. I, and observe that yn+1 is a linear combination of yn and j = 0, 1, . . , m— 1. , are functions of the relaxation parameters ω ; . ,. 10'); hence, consistency and stability conditions are easier satisfied. 10"); these formulas are explicit and require evaluation of the function / a t m points. Such formulas are, in fact, m-point Runge-Kutta formulas. 1). 2, § 1 SINGLE-STEP METHODS Next, we consider the formulas which arise when Newton-Raphson iteration is applied to the trapezoidal rule.

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