Parallel space-time solution strategy for the Navier Stokes equation

Biswajit Khara, Kumar Saurabh, Baskar Ganapathysubramanian (Iowa State University)
Robert Dyja (Czestochowa University of Technology)

We study solution strategies for solving time-dependent flow problems through a fully coupled space-time formulation. Such a method could be considered a member of the class of methods that attempts to exploit parallelism in both space as well as time when solving partial differential equations numerically. When developing such a methodology, the discrete problem presents an array of challenges for both linear as well as nonlinear equations. For example, the existence of an “advection” in time direction and the nonexistence of diffusion in the time direction renders the global Peclet number infinite. Thus proper stabilized methods need to devised for these problems. We formulate and implement a stabilized method based on the variational multiscale method (VMS). Furthermore, when solving nonlinear problems, in the absence of any “marching”, the lack of a good guess makes any quasi-Newton solve difficult to converge. We find a way around this issue by using an adaptive refinement strategy in space-time and seek coarse scale solution in earlier iterations and resolve smaller features progressively. To this end, we develop an aposteriori error indicator for the space-time stabilized variational problem. For specific application to flow problems using this method, we demonstrate examples using two benchmark problems in computational fluid dynamics: (i) the lid driven cavity and (ii) flow past a cylinder.