Exponential integrators for the incompressible Navier-Stokes equations.

Abstract

We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size.Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step.Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L 2 inner product. We examine this L 2 projection and an H 1 projection induced by the H 1 semi-inner product. The H 1 projection has an advantage over the L 2 projection in that it retains tangential Dirichlet boundary conditions for the flow. Both the H 1 and L 2 projections are solutions to saddle point problems that are efficiently solved by a preconditioned Uzawa algorithm.

On the use of exponential time integration methods in atmospheric models

“…This method also commonly appears in the literature as the 'exponentially fitted Euler' method, for example, Hochbruck et al (1998), Newman (2003. The second scheme we choose is the third-order EPI3:…”

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

“…Since the nonlinear convective term in the Burgers equation is shared by the Navier-Stokes equation, the presented results give a hint of the potential of the PEBK method as an efficient parallel solver in turbulent fluid dynamics, where nonlinearity plays a key role. The question remains how to treat the pressure in the incompressible Navier-Stokes equation, which enforces the incompressibility constraint on the velocity field (see for possible approaches in [15,43]). This will be explored in future work.…”

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

“…The question remains how to treat the pressure in the incompressible Navier-Stokes equation, which enforces the incompressibility constraint on the velocity field (see for possible approaches in [37,131]). This will be explored in Chapter 5.…”

“…In our approach, we reformulate the governing equations without the pressure variable by applying a divergence-free projection onto the Navier-Stokes equation. The reformulated equation is an ordinary differential equation that can be integrated with a Krylov subspace method for matrix exponential functions [131,146]. We adopted the EBK method, like in Chapter 4.…”

“…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 Krylovbased exponential integrators, see [37,131]. Other approaches for Krylov-based exponential integration in the context of fluid dynamics include the method of pseudo compressibility to find steady state solutions [146] and a method for the fully compressible Navier-Stokes equation [149].…”