Convergence of numerical ODE solvers and Lyapunov's theory of stability
| dc.creator | Viswanath, Divakar | |
| dc.date | 1999-04-23 | |
| dc.date.accessioned | 2026-07-07T05:28:48Z | |
| dc.date.available | 2026-07-07T05:28:48Z | |
| dc.description | For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and locally Lipshitz in $x$, and if necessary, several times continuously differentiable in $t$ and $x$. We associate a conditioning function $E(t)$ with each solution $x(t)$ which captures the accumulation of global error in a numerical approximation in the following sense: if $\tilde{x}(t;h)$ is an approximation derived from a single step method of time step $h$ and order $r$ then $\norm{\tilde{x}(t;h) - x(t)} < K(E(t)+ε)h^r$ for $0\leq t\leq T$, any $ε> 0$, sufficiently small $h$, and a constant $K>0$. Using techniques from the stability theory of differential equations, this paper gives conditions on $x(t)$ for $E(t)$ to be upper bounded linearly or by a constant for $t\geq 0$. More concretely, these techniques give constant or linear bounds on $E(t)$ when $x(t)$ is a trajectory of a dynamical system which falls into a stable, hyperbolic fixed point; or into a stable, hyperbolic cycle; or into a normally hyperbolic and contracting manifold with quasiperiodic flow on the manifold. | |
| dc.identifier | https://arxiv.org/abs/math/9904136 | |
| dc.identifier | http://arxiv.org/abs/math/9904136 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78400 | |
| dc.subject | Dynamical Systems | |
| dc.title | Convergence of numerical ODE solvers and Lyapunov's theory of stability | |
| dc.type | text |