New algorithms and models for computing fluid flow equations
Abstract: In this talk I will present several recent results in developing efficient and effective numerical methods for computing fluid flow equations. The first topic is on fast ensemble methods. Repeated computations of flow equations with varying parameters are commonly seen in many engineering and geophysical applications as an effort to deal with inherent uncertainties. These computations are generally treated as independent tasks. While parallel computing can save computational time in this setting, no savings are realized in terms of total computational cost. In this talk, I will present a new way to perform multiple simulations efficiently, in terms of both storage and computational cost. The proposed algorithm computes all realizations at one pass by adopting an ensemble time-stepping scheme, which results in the same coefficient matrix for all realizations. This reduces the problem of solving multiple linear systems to solving one linear system with multiple right-hand sides, for which efficient methods, e.g., block CG, block GMRES, have been developed to significantly reduce the computation cost. Then I will further talk about a new stabilization method for a family of second order timestepping methods for Naiver-Stokes equations and a new nonlocal turbulence model with fractional Laplacian modeling the superdiffusion in turbulence.