Multistep methods previous methods use information at xi to predict yi+1 multistep methods use information from xi-1, xi-2, etc. leads to better results

Simple 2nd order method to start Look at Heun method Euler predictor - O(h 2 ) Trapezoid corrector - O(h 3 )

To improve predictor, use third order Euler along with same old corrector iterated until converged

Can make one more modification - a predictor modifier Comes from error analysis of y i+1 0

So sequence is predict adjust prediction correct convereged? if not, correct again

Example of non-self starting Heun method Need two initial conditions

Non-self starting Heun method is O(h3) accurate in predictor and corrector Uses midpoint method for predictor (open) trapezoid method for corrector (closed) Try higher order methods

We have a number of options Newton-Cotes formulas open closed Adams formulas open (Adams-Bashforth) closed (Adams-Moulton)

Newton-Cotes approach - Open methods Given can write n is order of interpolating polynomial

n=1 (midpoint method) n=2 n=3

Newton-Cotes -- Closed methods General form Integral evaluated using closed Newton-Cotes formulas

n=1 n=2 (trapezoid rule) (Simpson’s 1/3 rule)

Adams formulas Open formulas - Adams Bashforth Example: How to get 2nd order Adams- Bashforth Taylor series rewrite

Use backwards difference for derivative and substitute

End up with second order Adams-Bashforth with error term Can do the same proceedure on other order Taylor series, other finite differences Get higher order Adams Bashforth

General form for Adams-Bashforth The b ’s are listed in your book Example: 4th order