To design efficient and accurate time integration schemes for numerically simulating incompressible viscous flows, such as those governed by Stokes or Navier-Stokes equations, the discretization and coupling of velocity and pressure need to be treated carefully for stability and consistency.  The gauge formulation introduces a gauge variable and an auxiliary field for the fluids equations, and the resulting system contains a coupled momentum equation and a kinematic equation with certain consistent boundary conditions. In this talk, we present an exponential time differencing multistep method for solving the gauge system with high-order temporal accuracy. In particular, the momentum equation is completely decoupled from the kinematic equation in the discrete level at each time step and is solved by explicit exponential time stepping schemes. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes flows. We also present a compact representation of the method for problems on rectangular domains, which makes FFT-based fast solvers available for the resulting fully discrete problem. Various numerical experiments are carried out to demonstrate the accuracy and stability of the proposed method. 

Dr. Zhu Wang obtained the Ph.D. in Department of Mathematics from Virginia Tech in 2012. He received the industrial postdoctoral fellowship from the Institute for Mathematics and its Applications, University of Minnesota from 2012 to 2014. After that, he joined University of South Carolina as an assistant professor in Department of Mathematics. Dr. Wang’s current research interests include scientific computing, reduced-order modeling, and developing mathematical tools for important industrial projects such as inverse photolithography, magneto-acoustic tomography with magnetic induction, etc.