Super-time-stepping Acceleration of Explicit Schemes for Parabolic Problems

The goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, at essentially no extra cost. The authors explain the basis of the method, namely stabilization via wisely chosen inner steps (stages), justify it for linear problems, and spell out how simple it is to incorporate in any explicit code for parabolic problems. Finally, we demonstrate its performance on the (linear) heat equation as well as on the (non-linear) classical Stefan problem, by comparing it with standard implicit schemes (employing SOR or Newton iterations). The results show that super-time-stepping is more efficient than the implicit schemes in that it runs at least as fast, it is of comparable or better accuracy, and it is, of course, much easier to program (and to parallelize for distributed computing).