DMS Applied Mathematics Seminar

Speaker: Sigal Gottlieb, University of Massachusetts, Dartmouth

Title: Developing high order, efficient, and stable time-evolution methods using a time-filtering approach.

Abstract: Time stepping methods are critical to the stability, accuracy, and efficiency of the numerical solution of partial differential equations. In many legacy codes, well-tested low-order time-stepping modules are difficult to change; however, their accuracy and efficiency properties may form a bottleneck. Time filtering has been used to enhance the order of accuracy (as well as other properties) of time-stepping methods in legacy codes. In this talk I will describe our recent work on time filtering methods for the Navier Stokes equations as well as other applications. A rigorous development of such methods requires an understanding of the effect of the modification of inputs and outputs on the accuracy, efficiency, and stability of the time-evolution method. In this talk, we show that time-filtering a given method can be seen as equivalent to generating a new general linear method (GLM). We use this GLM approach to develop an optimization routine that enabled us to find new time-filtering methods with high order and efficient linear stability properties. In addition, understanding the dynamics of the errors allows us to combine the time-filtering GLM methods with the error inhibiting approach to produce a third order A-stable method based on alternating time-filtering of implicit Euler method. I will present our new methods and show their performance when tested on sample problems.