Optimal time step control for the numerical solution of ordinary differential equations

In solving differential equations using a finite stepsize h, an error Eh(p+1) is generated at each step and propagates. This phenomenon is treated as a dynamical process, where h(t) is controlled to optimize a properly defined performance index. Applying the variational principle, optimal stepsize is found to be proportional to (E psi)(-1/p+1). where E is the error generation coefficient and psi is the adjoint function of the error variable. This means that conventional adaptive control strategies depending on local information (E) only are not optimal in general, except for two special cases. The theory is applied to three rest problems for two figures of merit. The method is compared with several conventional strategies.