Development of superconvergent hybridizable discontinuous Galerkin methods and mixed methods for Korteweg-de Vries type equations

University Of Massachusetts, Dartmouth, North Dartmouth MA

Investigators

Abstract

The project focuses on developing novel numerical methods for simulating the Korteweg-de Vries (KdV) type equations, that model phenomena in areas such as fluid mechanics, nonlinear optics, acoustics, and plasma physics. For example, the KdV equation has been used in the modeling of shallow water waves and the study of Tsunami waves. The new numerical tools developed under this project will provide scientists with a better understanding of theoretically unresolved issues on the mathematical properties of solutions to KdV type equations. Furthermore, the proposed project will provide accurate and efficient numerical algorithms for the simulation of nonlinear dispersive wave propagation in various applications. These proposed research topics will have a positive impact across the mathematical sciences and have significant applications in many scientific areas that rely on the study of non-linear phenomena. This project will involve undergraduate and graduate students and focus on involving student from groups traditionally underrepresented in the sciences. By working on the project, the students will benefit from novel ideas for new algorithm design, approaches for rigorous mathematical analysis, and advanced skills in implementation. The objective of the project is to devise and analyze the first superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized mixed methods for solving the KdV equations and their multidimensional generalizations. The proposed project includes a comprehensive coverage of new algorithm design that is backed up by solid analysis and made practical by efficient implementation. The P.I. proposes to carry out a detailed study of superconvergent HDG methods and hybridized mixed methods for KdV type problems in the following steps: First, the P.I. will develop novel HDG methods and hybridized mixed methods for stationary third-order linear equations, focusing on the discretization of the third-order differential operator. Superconvergence properties of the approximations will be computationally and analytically investigated. Second, the P.I. would like to solve the third-order KdV equations by using implicit schemes for time discretization to avoid extremely small time steps and developing new HDG methods and hybridized mixed methods for spatial discretization. Error analysis will be carried out, and superconvergence and conservativity properties will be studied. Third, the P.I. plans to extend these superconvergent methods to multidimensional KdV type equations such as the Kadomtsev-Petviashvili equation, and the hybridization technique will make the methods efficiently implementable in multiple dimensions.