5.6 Multistep Methods

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1 5.6 Multistep Methods 1

2 Motivation: Consider IVP: yy = ff(tt, yy), aa tt bb, yy(aa) = αα. To compute solution at tt ii+1, approximate solutions at mesh points tt 0, tt 1, tt 2, tt ii are already obtained. Since in general error yy(tt ii+1 ) ww ii+1 grows with respect to time tt, it then makes sense to use more previously computed approximate solution ww ii, ww ii 1, ww ii 2, when computing ww ii+1. Definition 5.14 An m-step multistep method for solving the IVP: yy = ff(tt, yy), aa tt bb, yy(aa) = αα has a difference equation for computing ww ii+1 at the mesh point tt ii+1 represented by: ww ii+1 = aa mm 1 ww ii + aa mm 2 ww ii aa 0 ww ii+1 mm +h[bb mm ff(tt ii+1, ww ii+1 ) + bb mm 1 ff(tt ii, ww ii ) + + bb 0 ff(tt ii+1 mm, ww ii+1 mm )] for ii = mm 1, mm,, NN 1, where h = (bb aa)/nn, the aa 0, aa 1,, aa mm 1 and bb 0, bb 1,, bb mm are constants, and the starting values ww 0 = αα, ww 1 = αα 1,, ww mm 1 = αα mm 1 are specified. 2

3 Remark. 1. When bb mm = 0, the method is called explicit; 2. When bb mm 0, the method is called implicit. Adams-Bashforth two-step explicit method. ww 0 = αα, ww 1 = αα 1 ww ii+1 = ww ii + h 2 [3ff(tt ii, ww ii ) ff(tt ii 1, ww ii 1 )] where ii = 1, 2, NN 1. Adams-Moulton two-step implicit method. ww 0 = αα, ww 1 = αα 1 ww ii+1 = ww ii + h 12 [5ff(tt ii+1, ww ii+1 ) + 8ff(tt ii, ww ii ) ff(tt ii 1, ww ii 1 )] where ii = 1, 2, NN 1. 3

4 Adams-Bashforth four-step explicit method. ww 0 = αα, ww 1 = αα 1, ww 2 = αα 2, ww 3 = αα 3 ww ii+1 = ww ii + h 24 [55ff(tt ii, ww ii ) 59ff(tt ii 1, ww ii 1 ) + 37ff(tt ii 2, ww ii 2 ) 9ff(tt ii 3, ww ii 3 )] where ii = 3, 4, NN 1. Adams-Moulton four-step implicit method. ww 0 = αα, ww 1 = αα 1, ww 2 = αα 2, ww 3 = αα 3 ww ii+1 = ww ii + h 720 [251ff(tt ii+1, ww ii+1 ) + 646ff(tt ii, ww ii ) 264ff(tt ii 1, ww ii 1 ) + 106ff(tt ii 2, ww ii 2 ) 19ff(tt ii 3, ww ii 3 )] where ii = 3, 4, NN 1. 4

5 Example 1. Solve the IVP yy = yy tt 2 + 1, 0 tt 2, yy(0) = 0.5 by Adams-Bashforth four-step explicit method and Adams-Moulton twostep implicit method respectively. Use the Runge-Kutta method of order four to get needed starting values for approximation and h = 0.2. Solution: By using Runge-Kutta method of order four: ww 0 = 0.5 yy(0.2) ww 1 = yy(0.4) ww 2 = yy(0.6) ww 3 =

6 Example. Derive Adams-Bashforth two-step explicit method for solving the IVP: yy = ff(tt, yy), aa tt bb, yy(aa) = αα. Integrate yy = ff(tt, yy) over [yy ii, yy ii+1 ] tt ii+1 tt ii+1 yy ii+1 yy ii = yy (tt)dddd = ff(tt, yy(tt))dddd tt ii Then form interpolating polynomial through tt ii, ff tt ii, yy(tt ii ), tt ii 1, ff tt ii 1, yy(tt ii 1 ) to approximate ff tt, yy(tt) and subsequently tt ii+1 ff(tt, yy(tt))dddd. tt ii tt ii 6

7 Definition 5.15 Local Truncation Error. If yy(tt) solves the IVP yy = ff(tt, yy), aa tt bb, yy(aa) = αα and ww ii+1 = aa mm 1 ww ii + aa mm 2 ww ii aa 0 ww ii+1 mm h[bb mm ff(tt ii+1, ww ii+1 ) + bb mm 1 ff(tt ii, ww ii ) + +bb 0 ff(tt ii+1 mm, ww ii+1 mm )], the local truncation error is: ττ ii+1 (h) = yy(tt ii+1) aa mm 1 yy(tt ii ) aa mm 2 yy(tt ii 1 ) aa 0 yy(tt ii+1 mm ) h [bb mm ff tt ii+1, yy(tt ii+1 ) + + bb 0 ff(tt ii+1 mm, yy(tt ii ))] for each ii = mm 1, mm,, NN 1. NOTE: the local truncation error of a m-step explicit step is OO(h mm ). the local truncation error of a m-step implicit step is OO(h mm+1 ). 7

8 Comparing m-step explicit step method vs. (m-1)-step implicit step method a) both have the same order of local truncation error, OO(h mm ). b) Implicit method usually has greater stability and smaller round-off errors. For example, local truncation error of Adams-Bashforth 3-step explicit method, ττ ii+1 (h) = 3 8 yy(4) (μμ ii )h 3. Local truncation error of Adams-Moulton 2-step implicit method, ττ ii+1 (h) = 1 24 yy(4) (ξξ ii )h 3. 8

9 Predictor-Corrector Method Motivation: (1) Solve the IVP yy = ee yy, 0 tt 0.25, yy(0) = 1 by the three-step Adams-Moulton method. Solution: The three-step Adams-Moulton method is ww ii+1 = ww ii + h 24 [9eeww ii ee ww ii 5ee ww ii 1 + ee ww ii 2] EEEE. (1) EEEE. (1) can be solved by Newton s method. However, this can be quite computationally expensive. (2) combine explicit and implicit methods. 9

10 4 th order Predictor-Corrector Method (we will combine 4 th order Runge-Kutta method + 4 th order 4-step explicit Adams-Bashforth method + 4 th order 3-step implicit Adams-Moulton method) Step 1: Use 4 th order Runge-Kutta method to compute ww 0, ww 1, ww 2 and ww 3. Step 2: For ii = 4, 5, NN (a) Predictor sub-step. Use 4 th order 4-step explicit Adams-Bashforth method to compute a predicated value ww ii+1,pp ww ii+1,pp = ww ii + h 24 [55ff(tt ii, ww ii ) 59ff(tt ii 1, ww ii 1 ) + 37ff(tt ii 2, ww ii 2 ) 9ff(tt ii 3, ww ii 3 )] 10

11 (b) Correction sub-step. Use 4 th order three-step Adams-Moulton implicit method to compute a correction ww ii+1 (the approximation at ii + 1 time step) ww ii+1 = ww ii + h 24 [9ff tt ii+1, ww ii+1,pp + 19ff(tt ii, ww ii ) 5ff(tt ii 1, ww ii 1 ) + ff(tt ii 2, ww ii 2 )] 11

multistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):

multistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP): MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem