Global errors of numerical ODE solvers and Lyapunov's theory of stability (Q2713134)

The exact solution of the initial value problem NEWLINE\[NEWLINEx'(t)= f(t,x),\quad x(0)= x_0,\quad t\geq 0NEWLINE\]NEWLINE cannot always be obtained by analytic methods. Numerical methods are developed to minimize the global error \(\|\widetilde x(t,h)- x(t)\|\). This paper introduces a conditioning function \(E(t)\), associated with the exact solution, which controls global errors. \(E(t)\) is defined as NEWLINE\[NEWLINEE(t)= \sup_v \Biggl\|\int^t_0 {\partial x(t)\over \partial x(s)} v(s) ds\Biggr\|,NEWLINE\]NEWLINE where the supremum is taken over all continuous functions with \(\|v(s)\|\leq 1\) for \(0\leq s\leq t\); \(v(s)\) is the direction of the discretization error made at time \(s\). \(E(t)\) is studied here for both linear and nonlinear systems.