Invariant curves for variable step size integrators
The behaviour of the numerical integration of the ordinary differential equation (1) \(\dot x=f(x)\), \(x\in \Omega \subset {\mathbb{R}}^ n\), (with a hyperbolic periodic solution) applying a one-step method with variable step size is investigated. It is shown that there exists an attractive (hyperbolic) invariant closed curve for the numerical method as in the case of one-step method with constant step size. If \(\phi\) is the integration method of order p and H is the step-size function it is shown that under general assumptions on H there is a function s(x,\(\epsilon\)) such that the set \(M_{\epsilon}:=\{(h,x)| h=\epsilon s(x,\epsilon)\), \(x\in \Omega \}\), is an attractive invariant manifold for the map \(\Psi:\;\left( \begin{matrix} h\\ x\end{matrix} \right)\to \left( \begin{matrix} \bar h\\ \bar x\end{matrix} \right)=\left( \begin{matrix} H(h,x,TOL)\\ \phi (h,x)\quad \quad \quad \end{matrix} \right),\) where \(\epsilon:=TOL^{1/p}\), \(TOL=| l(h,x)| /h\), \(\rho (h,x)=\phi (h,x)-\phi (h,x)\) with \(\rho\) (t,x) the solution of equation (1) with initial condition x and \(\phi\) (h,x) a map of an integration method (one- step method) of order p. So the step-size, asymptotically, does not depend on the previous step size but only depends on the position x and the parameter TOL (or \(\epsilon\)). Under time transformations which depend on the position x equation (1) takes the form \((2)\quad \frac{dx}{d\tau}=r(x)f(x),\) with r(x) a positive scalar function. With a proper choice of r(x) the variable step size method \(\phi\) of equation (1) is transformed into an integration method \(\phi\) of equation (2) with constant step size. So, general properties of constant step size methods carry over to variable step size methods. The results of one-step methods with constant step size may be applied to equation (2) and may be transferred to equation (1) where an integration method with variable step is applied.
scientific article; zbMATH DE number 4195074