A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method

The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale “stone forests” China and Madagascar. Mathematically, processes are modeled by a Stefan problem, which describes how motion phase-separating interface depends on local concentration gradients, coupled fluid flow. Simulating these problems is challenging, requiring evolution free whose normal derivatives an external field in ever-changing domain. Moreover, density differences domain induce self-generated convecting flows that further complicate numerical study processes. In this contribution, we present method for simulation problem scheme uses Immersed Boundary Smooth Extension solve bulk advection-diffusion equations complex, evolving geometry, ?-L provides stable boundary. We demonstrate 3rd-order temporal pointwise spatial convergence classical 2nd-order when Examples result high-Rayleigh number convection numerically studied, qualitatively reproduce complex morphologies observed recent experiments.