Abstract:
Finite element methods provide a robust, mature family of discretizations for partial differential equations, as they provide rigorous theory, general geometry, and fast solvers. Modern trends in computer architecture have tended to favor higher-order instances of these methods and the increased arithmetic intensity they provide. There is a need to combine these high-order finite element discretizations with high-order time stepping schemes. Commonly used temporal schemes like multi-step and diagonally implicit Runge-Kutta methods either lack A-stability or succumb to order reduction in the presence of stiffness. As a result, there is renewed interest in fully implicit Runge-Kutta schemes, especially those based on collocation (e.g., the so-called Gauss-Legendre and Radau IIA Runge-Kutta families). Such methods can provide a full suite of theoretically desirable properties, but they were largely written off as “primarily of academic interest” before 1980 on account of their high complexity.
In this talk, Dr. Kirby will describe recent progress on novel software and solvers for such methods. Working within the Irksome project, a high-level time stepping library built on top of the Firedrake project, Dr. Kirby was able to automate the construction of the complex, stage-coupled algebraic systems required by fully implicit methods. Thanks to Firedrake’s interface to the PETSc (Portable, Extensible Toolkit for Scientific Computation), it is possible to attack these systems with several kinds of effective preconditioned Newton-Krylov solvers. These new developments demonstrate that the theoretical potential of fully implicit methods can in fact be realized in practice, where they compare favorably to diagonally implicit methods in practical metrics (e.g., accuracy versus run-time) for some benchmark calculations in incompressible flow and phase field modeling.
Speaker’s Bio:
Dr. Robert Kirby is a Professor of Mathematics at Baylor University. He held prior faculty appointments at the University of Chicago and Texas Tech University after earning his Ph.D. at the University of Texas at Austin. His research has made contributions to both mathematical and computational aspects of numerical analysis, with work on fast algorithms and solvers for numerical partial differential equations as well as significant innovations in mathematical software for the automation of finite element methods. This latter work has been widely distributed through the FEniCS (Finite Element Computational Software) and Firedrake projects.
