Multiscale and Hybridizable Discontinuous Galerkin Methods for Dispersive Equations and Systems

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This project concentrates on the development of novel computational methods for efficiently solving dispersive equations and systems, including the time-independent Schrodinger equations and Korteweg-de Vries (KdV) type equations in multidimensional spaces and systems. Schrodinger equations play a central role in the study of quantum mechanical systems and are widely used in the simulation of quantum transport in nanoscale semiconductor devices. The proposed multiscale method for Schrodinger equations will have a positive impact in the study of quantum mechanics and great potential in applications to ultrafast, low consumption and high functionality nanoscale semiconductor devices. KdV type equations and systems have wide applications in various fields such as fluid mechanics, nonlinear optics, acoustics, plasma physics, and Bose-Einstein condensates. The proposed method for KdV type equations and systems will help understand theoretically unresolved issues and provide accurate and efficient numerical tools for simulation of nonlinear waves in applications. The proposed research includes the following topics, (1) development and analysis of multiscale discontinuous Galerkin (DG) methods for Schrodinger equations in 1D, system, and 2D for the simulation of nanoscale semiconductor structures on coarse meshes, (2) design and error analysis of hybridizable discontinuous Galerkin (HDG) methods for solving multidimensional KdV type equations and KdV type systems, and (3) design of IMEX HDG-DG schemes for efficiently solving KdV type nonlinear equations and systems. To efficiently resolve highly oscillatory solutions of Schrodinger equations on coarse meshes, the multiscale DG methods will incorporate the oscillatory nature of the solutions and thus the multiple scales into the non-polynomial basis functions. For KdV type equations in multi-dimensions and systems, the PI will devise new HDG methods, study their convergence and conservation properties, and combine them with other DG methods in IMEX scheme to achieve high-order solutions both in time and in space and avoid overly small time-step sizes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.