Section 1.8 Convergence, Consistency and Stability
The three concepts in the section's title are fundamental features to be owned by a numerical method. First we give a somehow abstract definition of what an ODE algorithm is:Definition 1.8.1.
By a fixed-stepsize (k+1)-steps algorithm Ah to solve a first order ODE ˙x=f(t,x) on Rn we mean a map Φh:Rn(k+1)→Rn. Here h is the fixed time-step. Given a set of initial k+1 points
x−k,…,x−1,x0,
the numerical solution of the ODE is defined recursively by
xn+1=Φh(xn,…,xn−k).
Definition 1.8.2.
A fixed-stepsize ODE algorithm Ah is convergent if, given any IVP
˙x=f(t,x),x(t0)=x0,
where f(t,x) satisfies the conditions for the existence and uniqueness of the solution, and any tf>t0 for which there exist a solution, the numerical solution {x1(h),…,xN(h)} converges to the exact solution x(t) as h→0, namely
lim
Definition 1.8.3.
A fixed-stepsize ODE algorithm A_h is consistent if, given any ODE \dot x=f(t,x) satisfying the conditions for the existence and uniqueness of the solution and any h small enough, the error of a single step of the algorithm goes to 0 as h\to0\text{.}Theorem 1.8.4.
Suppose that a one-step algorithm A_h has local error of order O(h^{k+1})\text{.} Then A_h has global error order O(h^{k})\text{.}Corollary 1.8.5.
For one-step ODE algorithms whose local error order is more than 1, consistency implies convergence.Definition 1.8.6.
A fixed-stepsize ODE algorithm A_h is stable if, given any ODE \dot x=f(t,x) satisfying the conditions for the existence and uniqueness of the solution and two points x_0,x'_0\text{,} the two corresponding numerical solutions \{x_1(h),\dots,x_N(h)\},\{x'_1(h),\dots,x'_N(h)\}\text{,} where Nh=t_f-t_0\text{,} are such that
\begin{equation*}
\|x_N(h)-x'_N(h)\|\leq C\|x_0-x'_0\|
\end{equation*}
for each small enough h and some constant C independent on h\text{.}Theorem 1.8.7.
A multi-step ODE algorithm is convergent if and only if it is stable and consistent.Definition 1.8.8.
A fixed-stepsize ODE algorithm A_h is linearly stable (or A-stable, or Asymptotically stable) if, given any linear ODE \dot x=Kx\text{,} for any initial condition x_0 we have that the corresponding numerical solution \{x_n(h)\}\text{,} for any fixed value of h\text{,} converges to 0 for n\to\inftyRemark 1.8.9.
This concept of stability is not necessarily about how close \{x_n(h)\} gets to the exact solution but rather about whether, as a discrete dynamical system, A_h is stable in a neighborhood of the 0 point. Indeed, the constant function x(t)=0 is always a solution of a linear ODE and so a linearly stable method A_h has the property that the orbit of any point converges asymptotically to this solution.
\begin{equation*}
\dot x = K x,\;K\in\bC
\end{equation*}
with Re(K)<0\text{.} In this case all solutions go to zero asymptotically and so any stable ODE algorithm will reproduce correctly this behavior. We will see in next section that, when Re(K)<0\text{,} the explicit Euler method is linearly stable only for h small enough, while the implicit method is so for every h