MILNE'S PREDICTOR CORRECTOR METHOD
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This document discusses Milne's predictor-corrector method for solving ordinary differential equations. Predictor-corrector methods use an explicit method (the predictor) to get an initial approximation, followed by iterations of an implicit method (the corrector) to refine the solution. Milne's method provides a built-in error estimate by comparing the predictor and corrector approximations, allowing for adaptive step size control. The document outlines the local truncation error and absolute stability properties of predictor-corrector methods.
MILNE'S PREDICTOR CORRECTOR METHOD
- 2. MILNE’S PREDICTOR-CORRECTOR METHOD -: CREATED BY :- ALAY MEHTA 141080106011 SHIVANI PATEL 141080106021 KAVIN RAVAL 141080106026 KUNTAL SONI 141080106028 SUBMITTED TO :- Prof . PARESH PATEL
- 3. MILNE’S PREDICTOR-CORRECTOR METHOD • Predictor Corrector Methods form the basis of the most successful codes for the solution of initial value problems of ordinary differential equations. • Briefly, these methods have been successful because they occur in naturally arising families covering a range of orders, they have reasonable stability properties, and they allow for easy error control via suitable step size/order changing policies and techniques.
- 4. MILNE’S PREDICTOR-CORRECTOR METHOD • Consider the implicit linear multistep method • A possible way of solving the nonlinear system (1) is via the fixed point iteration • where is given. This iteration will converge to the unique solution of (1) provided .
- 5. MILNE’S PREDICTOR-CORRECTOR METHOD • where L is the Lipschitz constant of f. Thus it converges for sufficiently small h. • The basic idea of predictor corrector methods is to compute the initial approximation idea of predictor corrector methods is to compute the initial approximation by an explicit linear multistep method (the predictor) and then run the iteration (2) for a predetermined number of steps. • The implicit method (1) is called the corrector.
- 6. Predictor Corrector Modes • We denote the corrector by (1) and the predictor by • And assume that the predictor and corrector have the same step number k.
- 7. Predictor Corrector Modes • There are various modes of predictor corrector methods. • Denoting by P the prediction, C one iteration of (2), and E an evaluation of f, these are denoted by P(EC) µE and defined by the boxed equations (5). • Note: There’s also a mode where the final evaluation is omitted, denoted by P(EC) µ , which, however, is rarely used in practice
- 8. Predictor Corrector Modes • Example: Suppose the predictor is Euler’s Method, and the corrector is the Backward Euler Method. Then the corresponding Predictor Corrector Method in PECE mode is given by Note: Thus predictor corrector methods constitute explicit methods in their own right. They do not constitute a particular way of implementing implicit methods (i.e., the corrector). Indeed, since they are explicit methods their stability properties may be quite distinct from those of the corrector.
- 9. The Local Truncation Error of Predictor Corrector Methods • The following is a simplified discussion of the local truncation error under the assumption that f is scalar valued, the orders of the predictor and corrector are the same (and both denoted by p), and the mode of the predictor corrector method is PECE. • The discussion is followed by a statement of more general results. • A full discussion of the subject can be found in the above mentioned book by Lambert. • For our purposes, by the local truncation error LTE of the PECE method we mean the difference
- 10. The Local Truncation Error of Predictor Corrector Methods • under the localizing assumption • We also recall the definition of the local truncation error of the predictor and corrector, i.e., using H.O.T. to denote higher order terms as usual:
- 11. The Local Truncation Error of Predictor Corrector Methods • And • Note: The local truncation error as defined here is identical to the error under the localizing assumption for explicit linear multistep methods, and differs only in higher order terms for implicit liner multistep methods. • To see this consider a general (not necessarily implicit) linear multistep method (1) and subtract (10) from (1).
- 12. The Local Truncation Error of Predictor Corrector Methods • If (1) is explicit the statement follows immediately, if it is implicit the statement follows upon applying the mean value theorem and absorbing the term hβk... in the H.O.T., exercise! Upon subtracting • from (9) and using (8) we obtain • Similarly, subtracting
- 13. The Local Truncation Error of Predictor Corrector Methods • from (10) we obtain • using (10) again
- 14. The Local Truncation Error of Predictor Corrector Methods • Absorbing in the higher order terms in the final expression of (14) we obtain. • The leading term of the local truncation error of the PECE method is therefore the same as that of the corrector alone. • In general, if P is a predictor of order p ∗ and C is a corrector of order p, the leading term local truncation error of the P(EC) µE method is
- 15. Absolute Stability of Predictor Corrector Methods • We proceed as we did for linear multistep methods. Consider the test equation y ′ = λy • where λ is complex. • We say that the predictor corrector method is absolutely stable for a given hλ if all solutions yn of the test equation (17) with step size h tend to zero as n tends to infinity. • Consider again a PECE method. For the test equation we have
- 16. Absolute Stability of Predictor Corrector Methods • thus we obtain • Similarly • Substituting (19) in (20), rearranging, and noting that ,yields
- 17. Absolute Stability of Predictor Corrector Methods • Despite its complicated appearance this is just a homogeneous constant coefficient linear difference equations whose solutions are linear combinations of powers of the roots of its characteristic polynomial. • Clearly, this is given by • Where • and
- 18. Absolute Stability of Predictor Corrector Methods
- 19. Predictor Corrector Methods • In general we obtain • Note: As for explicit linear multistep methods, the leading coefficient of the stability polynomial (which is 1) becomes small relative to some others as hλ tends to infinity. • As a consequence (at least) one of the roots must tend to infinity and thus become larger than 1. • Hence there can be no predictor corrector method whose region of absolute stability includes the entire left half plane. Indeed, no explicit convergent method of any type with an unbounded region of absolute stability has been found.
- 20. Milne’s Device • Crucial to any adaptive technique is an error estimate. • As we have seen several times, these are usually based on the comparison of two different approximations of the same thing. • Since predictor corrector methods compute several approximations of y (xn+k) they come with their built in error estimate. • The exploitation of this fact is called Milne’s Device. • Once again, we consider the case of a PECE method and assume that predictor and corrector have the same order p. • Modifications of the idea apply to other cases. • Recall from our discussion of the local truncation error that
- 21. Milne’s Device • And • Subtracting these equations yields • Solving for and plugging this into (28) yields
- 22. Milne’s Device • Ignoring the higher order terms this yields a computable estimate of the local error y(xn+k) − yn+k which can be used to decide whether or not a step is accepted, and also can be employed in the selection of the next step size (and order). • The linear multistep methods used most widely as predictors and correctors are the Adams-Bashforth and Adams-Moulton methods, respectively
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