On quasi-consistent integration by Nordsieck methods (Q1004018)

Quasi-consistent numerical integration of ordinary differential equations (ODEs) \(x'(t)=g(t,x(t))\) is available since 1976, but few use in practical applications has been done, mainly because of difficulties in the implementation such as the problem of quasi-consistent integration in a non-equidistant grid. The author studies the advantages of numerical integration by means of quasi-consistent Nordsieck formulae. He exploits the idea that the global error control can be as cheap as the local error control if the principal term of the local error dominates strongly over the remaining terms (the accumulated error does not exceed the principal term of the local error, being both terms of the same order with respect to the step-size) and considers Nordsieck-Adams-Moulton methods for an efficient global error control. The author also introduces implicitly extended Nordsieck methods to facilitate the local error control and studies the convergence of these methods. Moreover the paper ends with quite a few numerical examples using implicitly extended Nordsieck-Adams-Moulton formulae of orders from 3 to 7. One of the main theoretical achievements is the introduction and study of doubly quasi-consistent methods. Methods of this type accumulate the errors of two order higher than the local error of the numerical solution, meaning that the principal terms of local and global errors of a doubly quasi-consistent method coincide, and so the principal term of the local error is an asymptotically correct estimate of the true error. Unfortunately, as it is proved in the paper, Nordsieck schemes cannot be doubly quasi-consistent and although the idea of doubly quasi-consistent methods looks promising, and the theoretical background for this new global error evaluation is provided, no one has been found yet. Only fixed-step-size Nordsieck formulae are exploited in the paper, however using the idea that the local error can be a good estimate of the global one in super-quasi-consistent Nordsieck formulae, the usual local error control can be used to automatically provide a step-size for a user-supplied tolerance and an algorithm for suitable step-size selection is also provided.