|Special Guest Lectures|
|Towards time stable and high order accurate schemes for realistic|
|Postdoctoral Research Fellow|
|Johnston Hall 338
March 24, 2006 - 03:30 pm
For wave propagation problems, the computational domain is often large compared to the wavelengths, which means that waves have to travel long distances during long times. As a result, high order accurate time marching methods, as well as efficient high order spatially accurate schemes (at least 3rd order) are required. Such schemes, although they might be G-K-S stable (convergence to the true solution as delta x -> 0), may exhibit non-physical growth in time, for realistic mesh sizes. It is therefore important to device schemes, which do not allow a growth in time that is not called for by the differential equation. Such schemes are called strictly (or time) stable. We are particularly interested in efficient methods with a simple data structure that parallelize easily on structured grids. High order accurate finite difference methods fulfill these requirements. Traditionally, a successful marriage of high order accurate finite difference and strict stability was a complicated and highly problem dependent task, especially for realistic applications. The major breakthrough came with the construction (Kreiss et al., in 1974) of non-dissipative operators that satisfy a summation by parts (SBP) formulation, and later with the introduction of a specific procedure (Carpenter et al., in 1994) to impose boundary conditions as a penalty term, referred to as the Simultaneous Approximation Term (SAT) method. The combination of SBP and SAT naturally leads to strictly stable and high order accurate schemes for well-posed linear problems, on rectangular domains. During the last 10 years, the methodology has been extended to handle complex geometries and non-linear problems. In this talk I will introduce the original SBP and SAT concepts, and further discuss the status today and the focus on future applications. In particular I will discuss some recent developments towards time stable and accurate hybrid combinations of structured and unstructured SBP schemes, making use of the SAT method.
Ken Mattsson received his Ph.D. in scientific computing from Uppsala University in Sweden in 2003. He was a postdoctoral research fellow in the Department of Physics at the University of Florida in Gainesville. Mattsson is currently a postdoctoral research fellow at the Center for Turbulence Research (CTR) at Stanford University. His research interests are the following: time dependent wave propagation problems, high order finite difference methods, multi-physics and multi-solvers, nonlinear stellar pulsation, and turbulence research.