Toward High-Order Numerical Methods for Problems Involving Moving Interfaces, Jumps and Conserved Quantities

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The aim of the proposed program is to provide new numerical techniques for problems commonly tackled in the applied sciences and engineering. My main focus is to develop Cartesian grid methods for problems involving boundaries and interfaces. That is, the geometry of the problem is defined in an immersed setting by for example, a level set function.***The current proposal focuses on developing high-order Cartesian grid methods for: -1- the evolution of curves, surfaces, and arbitrary sets, -2- imposing interface jump conditions, and -3- developing active penalty methods for imposing boundary conditions. In addition, I am developing a side of my program linking conservation laws and structure-preserving discretizations. Here, I propose to discretize ODEs or PDEs by transforming (analytically) the equation into a conservation law. This conservation law can then be solved with exact discrete conservation by for example, the finite-volume method. These new schemes possess highly desirable long-term non-linear stability properties.***The ultimate goal of this 5-year program is to cover a wide swath from applications to theory. On the one hand, I plan to apply the techniques I am developing to tackle important problems from engineering e.g. fluid mechanics of single- (Euler eq.) and multi-phase flows (Navier-Stokes), elasticity, fluid-structure interaction, and electromagnetism. On the other hand, I am developing fundamental techniques e.g. evolution of arbitrary sets, exponential-resolution schemes, structure-preserving numerical methods. ***I plan on emphasizing 3D computations so as to be able to tackle realistic applications, and develop Cartesian grid methods that could in principle be general enough to be adapted to any type of mesh. The overarching objective of this proposal is to design numerical methods that are novel, general, and easy to implement with a focus on accuracy (high-order), stability (e.g. structure-preserving schemes), and efficiency (exponential-resolution methods in linear time).**