Discrete mechanics is a relatively recent technique to derive efficient and predictive space-time integrators that are faithful to the continuous world they are meant to simulate. As these integrators are based on mechanical principles, they inherit, at the computational level, the variational nature of most mechanical systems, from particle mechanics to continuum mechanics, with surprising efficiency and robustness, and with arbitrary accuracy. Our research goal is to further develop this technique and introduce an entirely new class of Eulerian-based variational integrators of partial differential equations. By design, our geometry-based time integrators will exhibit all of the classic hallmarks of variational integrators. Consequently our research, containing intertwined theoretical and practical components, is aimed at producing simple and predictive simulation tools for a wide range of phenomena in fluid dynamics, electromagnetism, magnetohydrodynamics, plasma physics, and reactive CFD.
While the idea of discretizing variational formulations of mechanics is standard for elliptic problems, only recently was it used to derive variational time-stepping algorithms for mechanical systems. We wish to leverage the large body of literature on Lagrangian and Hamiltonian principles in mechanics to develop discrete, Eulerian-based variational integrators and further extend the realm of geometric mechanical integrators developed over the past few years. The computational and mathematical foundations sought after, along with the resulting integrators, will provide fresh insight in understanding the fundamental role of discretization and numerics in Eulerian simulations. These new computational tools will consequently advance the state-of-the-art in a wide range of challenging physical problems such as magnetohydronamics, plasma physics, charged, and complex fluids. Our team, with existing strengths in multidisciplinary computational projects and academic-industry collaborations, will develop an innovative curriculum in mathematical computing to train future scientists in theoretical and computational matters. Besides providing a stepping stone for our long-term research goal of solving complex, multiscale physical phenomena, the synergetic mixture of mechanics and geometric objects in this project promises to be a rich source of fundamental and computationally important methods, as well as interdisciplinary interactions between physical sciences, mathematics, and information technology.
