RII Track-4:NSF: Construction of New Additive and Semi-Implicit General Linear Methods

The modeling of complex dynamical phenomena is critical to many fields of science and engineering. In particular, numerical simulations provide us with a greater understanding of natural processes and play an essential role in the design and optimization of engineering systems. In many cases, these simulations require the solution of time-dependent partial differential equations. Solving such equations is challenging due to the presence of wide-ranging spatial and temporal scales. There is therefore a continuing need for carefully designed time integration algorithms that allow modelers to simulate complex physical systems more efficiently and accurately. This EPSCoR RII Track-4 award provides a fellowship to an Assistant Professor at Tulane University and support for a graduate student. It will strengthen a collaboration between Tulane University and Los Alamos National Laboratory by enabling multiple visits to collaborate and interact with leading mathematicians and scientists. The project will develop new implicit general linear methods for solving stiff systems arising from the discretization of partial differential equations. These methods will be designed for both additively and nonlinearly partitioned problems, allowing the implicitness to be distributed in the most computationally efficient way. Moreover, the project will investigate differing types of implicitness, including fully-implicit methods that can leverage modern developments in preconditioning. Method design will draw on ideas from a recently introduced time integration framework that utilizes interpolating polynomials to greatly simplify the construction of multivalued integrators. Lastly, all the methods developed as part of this work will be validated and applied to solve a range of practical problems arising from the fields of plasmas, radiative transport, and fluid dynamics. 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.