My research focuses on developing algorithms for efficiently solving high-dimensional problems arising in the physical sciences. The kinetic equations of plasma physics, the Vlasov and Vlasov-Boltzmann equations, are specific interests of mine. Our toolbox for solving the Vlasov equation and its variants includes classical polynomial and Fourier spectral expansions of velocity space, discontinuous Galerkin methods on tensor-product bases, and dynamical low-rank methods that exploit low-rank separability of the six dimensional phase space. In addition to kinetic equations, I am interested in robust high-order methods for solving hyperbolic and mixed hyperbolic-parabolic conservation laws, again with a focus on the fluid equations of plasma physics: the magnetohydrodynamic and multi-fluid Maxwell systems of equations. Our toolbox for the fluid equations includes second-order high-resolution finite volume methods, high-order WENO finite volume methods, discontinuous Galerkin methods, and shock-capturing stabilizations of the latter.

Fluid equations arise as a particular stiff limit of kinetic equations, namely the collisional limit or the magnetized limit. This asymptotic structure, and the fact that many systems of physical interest are multi-scale, containing important dynamics at a wide range of temporal and spatial scales, leads to my interest in asymptotic-preserving methods, which can smoothly transition from solving the microscale kinetic dynamics to the macroscale fluid dynamics. Putting these ideas all together, I am interested in developing what we might call efficiency-preserving numerical methods: that is, methods which expend computational effort on resolving microscale dynamics as needed, but achieve the efficiency of a classical macroscale solver when that level of fidelity is justified by the dominant physics of the problem. Long-standing ideas in this direction include hp-adaptive finite element methods and domain decomposition techniques for separating physical space into "fluid-dominated" and "kinetic-dominated" regions. To these we add rank-adaptive dynamical low-rank methods and physics-informed dynamical low-rank decompositions, which exploit the dominant physics to propose an efficient low-rank ansatz.

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