On implicit algorithms of continuation method with applications to dynamic systems. (Q1426771)

This paper is concerned with the numerical solution of initial value problems for second order equations: \( \ddot{u}(t) = f (t, u(t), \dot{u}(t))\), \(u(t_0)=u_0 \in {\mathbb R}^n,\) \(\dot{u}(t_0)=\dot{u}_0 \in {\mathbb R}^n\). The authors propose to define the differential problem taking as new independent variable \( \lambda \) the arc length of the integral curve in the extended \((2n+1)\)-dimensional space \((t,u,\dot{u})\), so that \[ d \lambda = \sqrt{ 1 + v^T v + f^T(t,u,v) f(t,u,v)} \,dt \] and then to use some second order implicit discretization of the transformed equations. A fixed point iteration process is proposed to solve the implicit equations of the discretization and it is proved that, for \(f\) sufficiently smooth and a step size \( \Delta \lambda\) sufficiently small, the proposed iterative scheme converges to the unique solution of the implicit equations. Further, a modification of this iterative process is suggested to improve the rate of convergence to the unique solution. Finally, three examples are presented to test the behaviour of this technique of using the arc length as independent variable. In the first example a fixed step size is used along the numerical integration, whereas in the other examples the solution is computed with an automatic control of step size along the solution curve based on the well known Richardson extrapolation.