An Exponential Time Integrator for the Incompressible Navier--Stokes Equation

Gijs L. Kooij, Mike A. Botchev, Bernard J. Geurts
2018 SIAM Journal on Scientific Computing  
We present an exponential time integration method for the incompressible Navier-Stokes equation. An essential step in our procedure is the treatment of the pressure by applying a divergence-free projection to the momentum equation. The differential-algebraic equation for the discrete velocity and pressure is then reduced to a conventional ordinary differential equation that can be solved with the proposed exponential integrator. A promising feature of exponential time integration is its
more » ... l time parallelism within the Paraexp algorithm. We demonstrate that our approach leads to parallel speedup assuming negligible parallel communication. B685 An interesting alternative, the Paraexp method, has been introduced in [27] for the time-parallel solution of linear initial-value problems (IVPs). With Paraexp, the original problem is decoupled into nonhomogeneous and homogeneous subproblems. Parallel speedup is based on the observation that the homogeneous subproblems can be solved very fast with exponential integrators. These are time integration methods based on the exact integration of linear IVPs. Exponential integrators require efficient algorithms to compute the matrix exponential or its product with a vector. There are many methods for computing the matrix exponential. Particularly, Krylov subspace methods prove to be very suitable for the implementation of exponential integrators [16, 27, 48] . A detailed overview of exponential integrators is given in [36] . An extension of Paraexp to nonlinear IVPs has been introduced in [39]. Preliminary tests show that parallel speedup is realistic for the viscous Burgers equation, even for low viscosity coefficients. The approach in [39] is based on the exponential block Krylov (EBK) method [4] . A difference with the original formulation of Paraexp is that the exponential integrator, here the EBK method, is used to solve both the nonhomogeneous and the homogeneous subproblems. This unified approach is demonstrated to be effective, as the EBK method can also be very competitive in solving nonhomogeneous subproblems compared to conventional time integration methods. The application of exponential integrators to fluid dynamics is not straightforward. The incompressible Navier-Stokes equation, discretized in space, is a differentialalgebraic equation where a time derivative for the pressure is absent. The continuity equation acts as a constraint equation that imposes a divergence-free condition on the velocity field. The pressure is determined such that the velocity field remains divergence-free. Consequently, special care needs to be taken with exponential integration for the advancement of the pressure in time. One possible approach is to reformulate the governing equations by treating the pressure with a divergence-free projection of the Navier-Stokes equation. This reformulation gives a differential equation for the velocity field that can be solved with Krylov-based exponential integrators; see [17, 50] . Other approaches for Krylov-based exponential integration in the context of fluid dynamics include the method of pseudocompressibility to find steady state solutions [53] and a method for the fully compressible Navier-Stokes equation [55] . The EBK method and its potential parallelization with Paraexp have been demonstrated for the viscous Burgers' equation in [39] . Its application to incompressible flows is not trivial because of the structure of the governing equations. In this paper, we discuss how the EBK method can be extended to the incompressible Navier-Stokes equation. This also paves the way for parallel-in-time simulations of incompressible flows with Paraexp. We follow the approach of [17, 50] and treat the pressure by a divergence-free projection. The new time integration method is tested in several numerical experiments including the Taylor-Green vortex and a lid-driven cavity flow. We find that the EBK method can be applied successfully to incompressible flows and also in cases with rather low viscosity. Furthermore, we show that the EBK method can be used within the time-parallel framework outlined in [39] . We provide a simplified model to analyze how much speedup is feasible with the EBK method for parallelin-time simulations of the incompressible Navier-Stokes equation. This analysis indicates that with our current implementation of the EBK method a moderate parallel speedup can indeed be expected. This method could provide additional speedup on top of the speedup obtained with a conventional parallellization in space only. The paper is organized as follows. In section 2, we explain the basic principle of the EBK method and discuss how it can be applied to the incompressible Navier-Downloaded 09/12/18 to 130.89.12.106. Redistribution subject to SIAM license or copyright; see
doi:10.1137/17m1121950 fatcat:g6bjd4htzrg4zlxfh7q45cie4i