High Order Schemes: Bound Preserving, Moving Boundary, Stochastic Effects and Efficient Time Discretization

The project aims to develop efficient and high-precision numerical methods for solving partial differential equations in various important scientific and engineering applications, such as aerospace engineering, semiconductor device design, astrophysics, and biological applications. Even with today's fast supercomputers, it is still essential to design efficient and reliable algorithms to obtain accurate solutions to these applications where high precision can improve the safety and performance of those devices. These algorithms will make positive contributions to computer simulations of the complicated solution structure in these applications. The project will include workforce development for students from underrepresented groups in STEM. The project aims to investigate algorithm development, analysis, and application of high-order numerical methods, including discontinuous Galerkin (DG) finite element methods and finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes. The algorithms will be designed to solve linear and nonlinear convection-dominated partial differential equations (PDEs), emphasizing bound preserving, moving boundary, stochastic effects and efficient time discretization. Topics of the research investigations will include an inverse Lax-Wendroff procedure for numerical boundary conditions with moving boundaries and interfaces, mathematical properties and efficient solvers for forward-backward coupled PDE systems from traffic flow modeling, high order numerical methods for hysteretic flows, robust high order Lagrangian methods, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications including problems involving highly nonlinear constraints and one step Lax-Wendroff type time discretizations, problems with stiff source terms, high order DG schemes for stationary hyperbolic equations and radiative transfer equations, oscillation-free DG methods, and numerical solutions of stochastic differential equations. The research will provide guidelines for the algorithms' applicability and limitations while enhancing their accuracy, stability, and robustness. The research will include collaborations with engineers and other applied scientists to enable the efficient application of these new algorithms or new features in existing algorithms. 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.