Scientific Computing Multi-Step and Predictor-Corrector Methods

Overview One-Step Methods – use only info from previous step – Euler – Runge-Kutta Multistep Methods- use info from several prior steps – Adam Bashforth – Adam Moulton Method – Predictor-Corrector Method

Multi-Step Principle To solve We use an iteration scheme to find x i+1 in terms of previous values of x i, x i-1, x i-2, etc, and/or values of f i =f(t i, x i ), f i-1, f i-2, etc.

Multi-Step Principle The method comes from integrating the derivative to get x(t).

One-Step vs Multi-Step

Multi-Step Example Midpoint rule: Another weighted average rule:

Multi-Step General Form The general form for a multi-step method is The parameters a k and b k are determined by polynomial interpolation. If b m =0, the method is called explicit, as this formula gives x i+1 explicitly in terms of previously found values. If b m ≠0, the method is called implicit, as x i+1 appears on both sides of the equals sign.

Multi-Step Explicit Adams Method In this method we approximate the value of by interpolating f(t,x(t)) at the points (t i, x i ), (t i-1, x i-1 ), …, (t i+1-m, x i+1-m ). We then integrate this polynomial exactly to use in the formula for the next iterate:

Example: 3-Step Adams-Bashforth Want a formula of the type: We use the three previous values of (t i, x i ) for a Lagrange interpolating polynomial for f

Example: 3-Step Adams-Bashforth Then, After a change of variables: u=(t i+1 - t)/h we get

Example: 3-Step Adams-Bashforth Then, Now, Likewise,

Example: 3-Step Adams-Bashforth So, we get, Thus,

Example: 4-Step Adams-Bashforth

Implicit Multi-Step Methods Implicit multi-step methods use the value of x i+1 to find the value of x i+1. Of course, this is impossible if we do not know x i+1, so in practice we use an explicit method to approximate (predict) x i +1 and then use an implicit method to improve (correct) the value of x i+1. These methods again rely on polynomial interpolation approximation of f(t,x(t))

Adams-Moulton Implicit Methods Three-Point: Four Point:

Predictor-Corrector Methods The Predictor-Corrector technique uses an explicit scheme (like the Adams-Bashforth Method) to estimate the initial guess for x i+1 and then uses an implicit technique (like the Adams-Moulton Method) to correct x i+1.

Predictor-Corrector Example Adams third order Predictor-Corrector scheme: Use the Adams-Bashforth three point explicit scheme for the initial value. Use the Adams-Moulton three-point implicit method to correct.

Predictor-Corrector Example Consider Exact Solution Initial condition: x(0) = 1 Step size: h = 0.1 We will use the 3 Point Adams-Bashforth and 3 point Adams-Moulton. Both require 3 points to get started!

Predictor-Corrector Example From the 4th order Runge Kutta 3-point Adams-Bashforth Predictor Value:

Predictor-Corrector Example To correct, we need f(t 3, x 3 * ) 3-point Adams-Moulton Corrector Value:

The values for the Predictor-Corrector Scheme Predictor-Corrector Example

The predictor-corrector method produces a solution with nearly the same accuracy as the RK order 4 method. Generally, the n-step method will have truncation error of order at least n. Predictor-Corrector Example