Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Methods for Nonlinear Kinetics and Fluid Models

The primary goal of the project is the development of new computational methodologies for a wide range of transport-dominant systems in computational fluid dynamics. Methods with large time step sizes are still underdeveloped for kinetic and fluid applications. Theoretical foundations are yet to be established for quantifying the time stepping sizes allowed for stability.

There is great potential in further development of methodology for a moving mesh frame and application to moving boundaries and interfaces. This project will further state-of-art computational tools and theoretical analysis and aims to provide avenues for computational simulations that are currently intractable. The project involves training of graduate students through involvement in the research.

This project will develop a class of Eulerian-Lagrangian (EL) Discontinuous Galerkin (DG) approaches for linear and nonlinear transport-dominant partial differential equation models. The EL DG method is a generalization of the (semi-Lagrangian) SL DG method for linear advection problems, based on the design of a localized adjoint problem for the test function that approximately tracks characteristics.

Such features allow flexibility, especially for high dimensional and nonlinear problems, where characteristics are difficult to track. The errors occurred in approximating characteristics will be integrated in time by Runge-Kutta (RK) methods via the method-of-lines approach. This fully discrete scheme is termed "EL RK DG." When the characteristics are approximated well, the very restrictive CFL constraint in the RK DG framework can be relaxed, leading to CPU savings.

The EL RK DG method can be viewed as a general framework generalizing both the classical Eulerian RK DG formulation and the SL DG formulation. Thus, existing research on positivity preserving limiters, well-balanced treatments, asymptotic preserving properties, entropy stability, and error estimates on Eulerian RK DG methods can be potentially generalized to the EL RK DG framework.

A key goal is to establish large time-stepping size with nonlinear stability. The project will also explore generalization of the EL DG algorithm to a moving mesh reference frame for tracking material interfaces and moving boundaries.

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.