The objective of this project is to develop some robust and high accuracy numerical algorithms and related mathematical theory for Hamiltonian systems. The research efforts will be devoted to constructing "essentially" (accuracy is in a range of the computer round-off error) volume conserving (symplectic) and energy conserving algorithms. Some recent mathematical theory in spectral methods, finite element superconvergence, as well as discontinuous Galerkin methods will be employed in the project.
Phenomena in different scientific disciplines such as classical mechanics, molecular dynamics, hydrodynamics, electrodynamics, plasma physics, relativity, and astronomy, etc. can be described by Hamiltonian dynamical systems. The success of the project will impact science and engineering practices. This research will widen the body of knowledge in the scientific community on both mathematical theory and practical algorithmic design. The project contains solid multi- and interdisciplinary components and has wide application in scientific computing.
This project studies some highly effective numerical methods, such as polynomial spectral methods, finite volume methods and discontinuous Galerkin methods, in solving partial differential equations. The main concern is the superconvergence phenomenon. In scientific computing, convergence rate at some special points is sometimes higher than the best possible global rate, these points are called superconvergence points and the phenomenon is called superconvergence phenomenon. The superconvegence phenomenon is well understood for the traditional finite element methods and researchers in this old field have accumulated a vast literature in the past 40 years. As a comparison, the relative study for spectral methods, finite volume methods, and discontinuous Galerkin methods are lacking. In this project, the PI developed some systematic way to search for superconvergence points of polynomial spectral collocation methods, a class of finite volume methods, and the local discontinuous Galerkin methods. As a result, many new superconvergence points have been discovered and reported for the first time in the literature. These discoveries will provide guidance to numerical analysits and practitioners to access computational data more accurately, to build a posteriori error estimators, and to construct adaptive procedure for aforementioned numerical methods. Scientists working on scientific and engineering computation would benefit from the results of this project. The project contains training of Ph.D. students at Wayne State University.