Development of Discontinuous Galerkin Methods for Kinetic Transport Models and Control Problems with State Constraints

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The objective of this project is to develop and analyze novel discontinuous Galerkin (DG) methods for solving partial differential equations arising from various application areas. The DG method is a class of finite element methods using completely discontinuous piecewise polynomial space for the numerical solution and the test functions. Those robust, compact, locally conservative methods can treat arbitrarily unstructured meshes and are ideal for hp-adaptive strategies. The good properties of the scheme call for further research in areas that are traditionally not solved by DG methods. In this grant proposal, the PI plans to conduct research in the following directions: (1) a positivity-preserving DG method for solving the kinetic equations, including the Boltzmann equations and Vlasov equations, (2) application of the proposed method to solar cell/semiconductor device simulations and plasma physics, (3) a novel DG solver for the Hamilton-Jacobi equations and its applications in control problems with state constraints. The proposed activity lies between algorithm development, analysis and applications. Developing robust, high-order accurate, cost-efficient numerical algorithms for kinetic models and control problem is very challenging, not only because of the high dimensionality of such models, but also because of the fact that a deep understanding of the underlying physics is required. The eventual goal is to produce solvers that are computationally efficient and suit the need for applications. The PI's work arises from the computational demand of real world applications. Many ideas developed in this proposal will have straightforward applications and impacts in semiconductor device simulations, high-efficiency fuel cell modeling, control problems and plasma physics. The PI actively interacts with students and faculty members in mathematics, physics, electrical engineering and chemistry departments. In addition, the PI will integrate the project with the training of graduate students in order to communicate in a broader context.