Moore’s law and the Saffman–Taylor instability

Abstract

Ten years ago Hou, Lowengrub and Shelley [4] published a state-of-the-art boundary integral simulation of a classical viscous fingering problem, the Saffman–Taylor instability [12]. In terms of complexity and level of detail, those computations [4] are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature. Since 1994, the computational power of a standard workstation has increased a hundredfold as predicted by Moore s law [7]. The purpose of this Note is to consider Moore s law and its consequences in computational science, and in particular, its impact on studying the Saffman–Taylor instability. We illustrate Moore s law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date. Viscous fingering is one of the fundamental interfacial instabilities in fluid dynamics: Perturbations to an expanding circular air bubble displacing a viscous fluid in a thin gap flow device become unstable, resulting in intricate densely branched interfacial patterns. The viscous fingering problem is governed by the three-dimensional incompressible Navier–Stokes equations with moving free boundaries at the fluid/ air interface. Without simplification, this problem is computationally intractable and will remain so for the foreseeable future. The simulations in this Note build upon decades of advances in mathematical modeling and numerical methods: (i) The fluid dynamics in a thin gap is reduced from the threedimensional Navier–Stokes equations by asymptotic analysis to a Darcy s law [12]