Abstract:  High-order immersed methods are a class of PDE discretizations that simulate complex geometries on a background Cartesian grid while maintaining high-order accuracy in both space and time. These schemes are particularly useful for fluid simulations with moving or deforming geometries, for which the cost of maintaining a moving body-fitted mesh can become prohibitive. This talk provides an overview of the current landscape of high-order immersed methods, as well as recent work within the MIT van Rees Lab to develop immersed interface methods (IIMs) for 3D PDE simulations with moving boundaries. We discuss spatial discretizations that combine standard finite difference schemes with a weighted least-squares reconstructions of the PDE solution near immersed boundaries, and assess the stability of these methods when applied to hyperbolic PDEs. We also discuss the issue of "freshly cleared cells" in simulations with moving boundaries, and demonstrate a method that maintains the high-order accuracy of explicit Runge-Kutta time integrators even in the presence of moving boundaries. Finally, we demonstrate a high-performance implementation of these discretizations within MURPHY, a scalable software framework for  3D multiresolution grids with wavelet-based adaptivity. Through 3D simulations we assess the stability and convergence of our IIM when applied to advection-diffusion simulations with moving boundaries, an important step on the path to a high-order solver for the incompressible Navier-Stokes equations.