High Order Schemes: Robustness, Efficiency, and Stochastic Effects

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This project concerns algorithm design and analysis of efficient, highly accurate numerical methods for solving partial differential equations. Such equations are used in simulation of systems arising in diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biology. Even with today's fast computers, efficient computational solution of partial differential equations remains a challenge, and it is essential to design improved algorithms that can be used to obtain accurate solutions in these application models. The research aims to produce a suite of powerful computational tools suitable for computer simulations of the complicated solution structure in these applications. The project provides training for a graduate student through involvement in the research.This project conducts research in 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, for solving linear and nonlinear convection-dominated partial differential equations, emphasizing scheme robustness, efficiency, and the treatment of stochastic effects. The project focuses on algorithm development and analysis. Topics of investigation include a new class of multi-resolution WENO schemes with increasingly higher order of accuracy, an inverse Lax-Wendroff procedure for high-order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications, entropy stable DG methods, optimal convergence and superconvergence analysis of DG methods, numerical solutions of stochastic differential equations, and the study of modeling, analysis, and simulation for traffic flow and air pollution. Applications motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations and to enhance their accuracy, stability, and robustness; and collaborations with engineers and other applied scientists will enable the efficient application of these new 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.