Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Systems with Compact Stencils

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The main objective of this project is to systematically develop a novel class of efficient and high order accurate Runge–Kutta (RK) discontinuous Galerkin (DG) methods for convection-dominated problems and the related applications. The new methods feature improved compactness and local structures. They are expected to be more suitable for parallel computing and implicit time marching in computational fluid dynamics simulation. They have potential applications in diverse areas such as meteorology, oceanography, gas dynamics, aircraft design, hydraulic engineering, oil recovery simulation, and so on. The project will also provide research opportunities for graduate and/or undergraduate students interested in computational mathematics and benefit curriculum development in the PI’s department. In more detail, the PI will investigate a novel approach to reduce the stencil size of the traditional RKDG methods, which typically grows with the number of RK stages. The resulting new methods are referred to as the compact RKDG methods. A comprehensive study of the methods will be carried out in the following directions. Firstly, high order compact RKDG methods will be designed for nonlinear hyperbolic conservation laws. Techniques for oscillation control, implicit time marching, and parallel computing will be investigated. Secondly, a rigorous theoretical framework for convergence, stability, and error analysis of the compact RKDG methods will be established. Thirdly, numerical techniques to preserve the solution bounds and investigate their applications to nonlinear hyperbolic systems in multidimensions will be developed. Finally, in addition to purely convection equations, the methods will be extended to convection-diffusion problems for simulation of viscous flow.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.