Lecture 4 - 4 Stiness and Stability In addition to having a...

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4 Stiffness and Stability In addition to having a stable problem , i.e., a problem for which small changes in the initial conditions elicit only small changes in the solution, there are two basic notions of numerical stability. The first notion of stability is concerned with the behavior of the numerical solution for a fixed value t > 0 as h 0. Definition 4.1 A numerical IVP solver is zero stable if small perturbations in the initial conditions do not cause the numerical approximation to diverge away from the true solution provided the true solution of the initial value problem is bounded. For a consistent s -step method one can show that the notion of stability and the fact that its characteristic polynomial ρ satisfies the root condition are equivalent. Therefore, as mentioned earlier, for an s -step method we have convergence ⇐⇒ consistent & stable . This concept of stability also plays an important role in determining the global truncation error. In fact, for a convergent (consistent and stable) method the local truncation errors add up as expected, i.e., a convergent s -step method with O ( h p +1 ) local truncation error has a global error of order O ( h p ). 4.1 Linear Stability Analysis A second notion of stability is concerned with the behavior of the solution as t → ∞ with a fixed stepsize h . This notion of stability is often referred to as absolute stability , and it is important when dealing with stiff ODEs. An absolutely stable numerical method is one for which the numerical solution of a stable problem behaves also in this controlled fashion. A Model Problem: For λ R we consider the family of scalar linear initial value problems (the discussion in the book by Iserles uses the analogous system case) y ( t ) = λy ( t ) , t [0 , T ] , y (0) = y 0 . The solution of this problem is given by y ( t ) = y 0 e λt . Now we take the same differential equation, but with perturbed initial condition y δ (0) = y 0 + δ. Then the general solution still is y δ ( t ) = ce λt . However, the initial condition now implies y δ ( t ) = ( y 0 + δ ) e λt . Therefore, if λ 0, a small change in the initial condition causes only a small change in the solution and therefore the problem is a stable problem . However, if λ > 0, then 63
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large changes in the solution will occur (even for small perturbations of the initial condition), and the problem is unstable . Therefore, when studying the stability of numerical methods we will consider the model problem for λ 0 only. Even though we will study only stability with respect to the model problem, it can be shown that the results of this analysis also apply to other linear (and some nonlinear) problems.
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