lecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 38: Linear Multistep Methods: Absolute Stability, Part II

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1 lecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 3: Linear Multistep Methods: Absolute Stability, Part II 5.7 Linear ultistep ethods: absolute stability At this point, it ay well see that we have a coplete theory for linear ultistep ethods. With an understanding of truncation error and zero stability, the convergence of any ethod can be easily understood. However, one further wrinkle reains. (Perhaps you expected this: thus far the b j coefficients have played no role in our stability analysis!) Up to this point, our convergence theory addresses the case where h!. Methods differ significantly in how sall h ust be before one observes this convergent regie. For h too large, exponential errors that reseble those seen for zero-unstable ethods can eerge for rather benign-looking probles and for soe ODEs and ethods, the restriction iposed on h to avoid such behavior can be severe. To understand this proble, we need to consider how the nuerical ethod behaves on a less trivial canonical odel proble. Now consider the odel proble x (t) =lx(t), x() =x for soe fixed l C, which has the exact solution x(t) =e tl x. In those cases where the real part of l is negative (i.e., l is in the open left half of the coplex plane), we have x(t)! as t!. For a fixed step size h >, will a linear ultistep ethod iic this behavior? The explicit Euler ethod applied to this equation takes the for For an elaboration of any details described here, see Chapter of Süli and Mayers. x k+ = x k + hf k = x k + hlx k =( + hl)x k. Hence, this recursion has the general solution x k =( + hl) k x. Under what conditions will x k!? Clearly we need + hl < ; this condition is ore easily interpreted by writing + hl = hl, where that latter expression is siply the distance of hl fro in the coplex plane. Hence + hl < provided hl is located strictly in the interior of the disk of radius in the coplex plane, centered at. This is the stability region for the explicit Euler ethod, shown in the plot on the next page. Now consider the backward (iplicit) Euler ethod for this sae odel proble: x k+ = x k + hf k+ = x k + hlx k+.

2 5.5 Forward Euler.5 Backward Euler x k+ = x k + hf k x k+ = x k + hf k+ Solve this equation for x k+ to obtain x k+ = hl x k, Figure 5.9: Stability regions for the forward and backward Euler ethod. If hl is contained within the blue region, then the approxiate solution {x k } to x (t) =lx(t) will converge, x k!, as k!. fro which it follows that x k =( hl) k x. Thus x k! provided hl >, i.e., hl ust be ore than a distance of away fro in the coplex plane. As illustrated in the plot on the next page, the backward Euler ethod has a uch larger stability region than the explicit Euler ethod. In fact, the entire left half of the coplex plane is contained in the stability region for the iplicit ethod. Since h >, for any value of l with negative real part, the backward Euler ethod will produce decaying solutions that qualitatively iic the exact solution. If hl falls within the stability region for a ethod, we say that the ethod is absolutely stable for that value of hl. Figure 5.9 shows the stability regions for the forward and backward Euler ethods. The blue region shows values of lh in the coplex plane for which the ethod is absolutely stable. (For the backward Euler ethod, this regions extends throughout the coplex plane, beyond the range of the plot.) A general linear ultistep ethod j= a j x k+j = h j= b j f k+j

3 applied to x (t) =lx, x() =x reduces to j= which can be rearranged as a j x k+j = hl j= j= b j x k+j, (a j hlb j )x k+j. This expression closely resebles the forula we analyzed when assessing the zero stability of linear ultistep ethods, except that now we have the hlb j ters. The new equation is also a linear constant-coefficient recurrence relation, so just as before we can assue that it has solutions of the for x k = g k for constant g. The values of g C for which such x k will be solutions to the recurrence are the roots of the stability polynoial which can be written as j= (a j hlb j )z j, r(z) hls(z) =, where r is the characteristic polynoial, r(z) = a j z j j= and s(z) = b j z j. j= Thus for a fixed hl, there will be solutions of the for g k j for the roots g,...,g of the stability polynoial. If these roots are all distinct, then for any initial data x,...,x we can find constants c,...,c such that x k = c j g k j. j= For a given value hl, we have x k! provided that g j < for all j =,...,. If that condition is et, we say that the linear ultistep ethod is absolutely stable for that value of hl. In the next section, we will describe how linear systes of differential equations, x (t) =Ax(t), can give rise, through an eigenvalue decoposition of A, to the scalar proble x (t) =lx(t) with coplex values of the eigenvalue l (even if A is real). This explains our interest in values of hl C.

4 7 We can now add a condition to our growing list of requireents to look for when assessing the quality of a linear ultistep ethod. We seek linear ultistep ethods with the following properties: high order truncation error; zero stability; absolute stability region that contains as uch of the left half of the coplex plane as possible. Those ethods for which the stability region contains the entire left half plane are distinguished, as they will produce, for any value of h, exponentially decaying nuerical solutions to linear probles that have exponentially decaying true solutions, i.e., when Re l <. Definition 5.. A linear ultistep ethod is A-stable provided that its stability region contains the entire left half of the coplex plane. Figure 5. shows the stability regions for two Adas Bashforth and Adas Moulton ethods. Notice two trends in these plots: () the iplicit Adas Moulton ethods have a larger stability region than the explicit Adas Bashforth ethods of the sae order; () as the order of the ethod increases, the stability region gets saller. Figure 5. shows the stability regions for a class of iplicit integrators called backward difference ethods. The -step backward difference ethod is siply the trapezoid ethod described earlier. All four of these ethods have contain the entire negative axis within their stability region, which will ake these ethods very effective for iportant systes of differential equations we will discuss in the next lecture. How does one draw plots of the sort shown here? We take the second order Adas Bashforth ethod x k+ x k+ = h( 3 f k+ f k) as an exaple. Apply this rule to x (t) = f (t, x(t)) = lx(t) to obtain x k+ x k+ = lh( 3 x k+ x k), with which we associate the stability polynoial z ( + 3 lh)z + lh =. Any point lh C on the boundary of the stability region ust be one for which the stability polynoial has a root z with z =. We can rearrange the stability polynoial to give lh = z 3 z z.

5 .5 Second-Order Adas Bashforth.5 Second-Order Adas Moulton x k+ x k+ = h 3 f k f k x k+ x k = h( f k + f k+ ).5 Fourth-Order Adas Bashforth.5 Fourth-Order Adas Moulton x k+ x k+3 = h 55 f k+3 59 f k f k+ 9 f k x k+3 x k+ = h 9 f k f k+ 5 f k+ + f k For general ethods, this expression takes the for (5.) lh = j= a jz j j= b jz j, To deterine the boundary of the stability region, we saple this forula for all z C with z =, i.e., we trace out the iage for z = e iq, q [, p). This curve will divide the coplex plane into stable and unstable regions, which can be distinguished by testing the roots of the stability polynoial for lh within each of those regions. We illustrate this process for the fourth order Adas Bashforth schee. The curve described in the last paragraph is shown in Figure 5.; it divides the coplex plane into regions where the stability Figure 5.: Stability regions for the second-order and fourth-order Adas Bashforth (explicit) and Adas Moulton (iplicit) ethods. If hl is contained within the blue region, then the approxiate solution {x k } to x (t) =lx(t) will converge, x k!, as k!.

6 9 -Step Backward Difference Method (Trapezoid) Step Backward Difference Method - - x k+ x k = h( f k + f k+ ) 3x k+ x k+ + x k = hf k+ 3-Step Backward Difference Method -Step Backward Difference Method x k+3 x k+ + 9x k+ x k = hf k+3 5x k+ x k+3 + 3x k+ x k+ + 3x k = hf k+ polynoial has an equal nubers of roots larger than in agnitude. As denoted by the nubers on the plot: outside the curve there is one root larger than one; within the rightost lobes of this curve, two roots are larger than one; within the leftost region, no roots are larger than one in agnitude. The latter is the stable region, which is colored blue in the botto-left plot in Figure 5.. Figure 5.: Stability regions for four (iplicit) backward difference ethods. If hl is contained within the blue region, then the approxiate solution {x k } to x (t) =lx(t) will converge, x k!, as k!.

7 .5 Figure 5.: The curve traced out by j= a je jiq / j= b je jiq for q [, p). The nubers reveal the nuber of roots of the stability polynoial that have agnitude larger than one. The stability region is the region bounded by this curve for which all the roots of the stability polynoial are less than one in agnitude