Title: IMEX-RK methods

Speaker: Alex Kanevsky, Brown University, von Neumann Fellowship candidate

Date/Time: Tuesday, March 8, 2005, 9:00 - 10:00 a.m.

Location: Building 980, Room 95 (Sandia NM), Building 915, Room 133W (Sandia CA)

Brief Abstract: We are interested in removing one of the main limitations of high-order fully explicit time integration schemes for problems which suffer from high levels of geometry-induced stiffness: the severe stability-based time-step restriction. Geometry-induced stiffness, or scale-separation stiffness, is a result of attempting to simultaneously simulate a system that has geometric features of drastically varying scales. We implement high-order Implicit-Explicit Runge-Kutta (IMEX-RK) methods to overcome geometry-induced stiffness in various linear and nonlinear problems on unstructured grids, by solving the non-stiff portions of the domain using explicit methods, and solving the more expensive stiff portions using implicit methods. We follow the method of lines approach, and discretize space using a nodal Discontinuous Galerkin Spectral Element method.

Surprisingly, IMEX-RK methods are not only more efficient than ERK methods for sufficient levels of stiffness, but are also more accurate when nonconsistent filters are used. We identify the fundamental mechanism by which filtering can become an additive process. We demonstrate analytically and numerically that filtering a numerical approximation which is at or near steady-state, for which the time step is small, or a combination thereof will result in a growing filtering-in-time error which will erase modes from the Fourier or polynomial approximation when nonconsistent filters are used, and suggest the development and implementation of time-dependent, time-consistent filters to control this error.

This is joint work with David Gottlieb and Jan S. Hesthaven (Brown University) and Mark H. Carpenter (NASA Langley).

CSRI POC: Scott Collis, (505) 284-1123