Beth Wingate

Adam Larios, Mark Petersen, Edriss S. Titi et al.

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $α>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.

Terry Haut, Beth Wingate

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable.

Adam Peddle, Terry Haut, Beth Wingate

A variant of the Parareal method for highly oscillatory systems of PDEs was proposed by Haut and Wingate (2014). In that work they proved superlinear conver- gence of the method in the limit of infinite time scale separation. Their coarse solver features a coordinate transformation and a fast-wave averag- ing method inspired by analysis of multiple scales PDEs and is integrated using an HMM-type method. However, for many physical applications the timescale separation is finite, not infinite. In this paper we prove con- vergence for finite timescale separaration by extending the error bound on the coarse propagator to this case. We show that convergence requires the solution of an optimization problem that involves the averaging win- dow interval, the time step, and the parameters in the problem. We also propose a method for choosing the averaging window relative to the time step based as a function of the finite frequencies inherent in the problem.