Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
For researchers and practitioners in numerical analysis, it highlights how discrete-specific thinking can overcome long-standing challenges more efficiently than continuous-inspired approaches.
The paper surveys three major difficulties in numerical ODE solution and presents novel, simpler solutions that avoid the pitfalls of directly mimicking continuous methods, showing that discrete-specific tools can be more effective.
On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.