Modelling multiphase matter: from microparticles to mega-icebergs

The world is multiphase. Water and ice, rock and lava, nucleus and cytoplasm. How can we model these systems, and simulate them efficiently? I'll start with three examples from my research, boat dynamics in dead water, melting icebergs in salty oceans, and phase-separating polymers in microfluidic experiments. The same patterns recur. A seemingly simple partition into PDEs and boundary conditions can miss the narrow transition between phases. This diffuse interface in turn motivates a host of new numerical schemes. The immersed-boundary method, volume-penalty techniques, and phase-field models are a handful of examples. The bulk of my talk will discuss the mathematical tools we need to understand and improve these methods. Signed-distance coordinates give a straightforward vector calculus around arbitrary submanifolds, and multiple scales matched asymptotics describes the resulting solutions to arbitrary order. I'll also discuss efficient spectral discretisations of these schemes, emphasising how our mathematical tools can improve accuracy and alleviate stiffness, before concluding with some bigger questions for multiphase methods.