Use the solution of the above equation with the a

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Unformatted text preview: this case, the solution varies rapidly until the transient exponential term dies out. It then settles onto the slowly varying polynomial part of the solution as shown in Figure 2.2.1. The rapidly varying part of the solution is called the \inner solution." It is non-trivial in a narrow \initial layer" near = 0. The slowly varying part of the solution dominates for most of the time and is called the \reduced" or \outer" solution. Were we to solve this problem by Euler's method, we would introduce spurious oscillations unless the absolute stability condition (2.1.16) were satis ed. This would require 2 or 2 . When is large this can be quite restrictive. Small perturbations to the outer solution quickly decay. Since perturbations arise naturally in numerical computation, Euler's method will require small step sizes to satisfy stability condition (2.1.16) even when the solution is varying slowly (Figure 2.2.2). t h h = 19 1.4 1.2 1 y 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 Figure 2.2.1: Solution of = ; ( ; 2)+2 (solid) illustrating inner (dashed) and outer (dash-dot) solutions. y 0 y t t As shown, a small perturbation to the outer solution is introduced at = 0 4. As noted, the exact solution to this perturbed solution quickly decays to the outer solution. The solution by Euler's method follows the steep initial slope of the perturbed solution and overshoots the outer solution. These overshoots and undershoots will continue in subsequent time steps as illustrated by Example 2.1.4 Problems of this type are called \sti ." They arise in applications where phenomena occur on vastly di erent time scales. Typical examples involve chemical kinetics, optimal control, and electrical circuits. There are many mathematical de nitions of sti ness and the one that we will use follows. t : De nition 2.2.1. A problem is sti in an interval if the step size needed for absolute stability is much smaller than the step size required for accuracy. 20 1 0.9 0.8 0.7 y 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 Figure 2.2.2: Euler solution (dash-dot) due to a perturbation (dashed) of = introduced at time = 0 4. y t 1 2 t (solid) : Sti ness not only depends on the di erential equation under consideration, but the interval of interest, the accuracy criteria, and the region of absolute stability of a numerical method. For nonlinear systems y = f ( y) 0 t sti ness will be related to the magnitudes of the eigenvalues of the Jacobian fy ( y). These eigenvalues may have vastly di erent sizes that vary as a function of . Thus, detecting sti ness can be a signi cant problem in itself. t t In order to solve sti problems we'll need a numerical method with a weaker stability restriction than Euler's method. \Implicit methods" provide the currently accepted approach and we'll begin with the simplest implicit method, the backward (or implicit) 21 Euler method. Once again, consider the scalar IVP y 0 =( ) fty t> 0 y (...
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