An exponential time-integrator scheme for steady and unsteady inviscid flows

Abstract

An exponential time-integrator scheme of second-order accuracy based on the predictorcorrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2).For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

Nonlinear p-multigrid preconditioner for implicit time integration of compressible Navier--Stokes equations

“…We are aware that storing the element-Jacobi smoother of the finest p-sublevel is very overwhelming for such hardware, especially when the polynomial degree is high. For JFNK-pMG on GPU, we would like to explore the feasibility of using a first order exponential time integrator [46] as the smoother instead of the EJ smoother and the matrix-based smoother. The overall RAM usage can potentially be minimised without the presence of an element-Jacobi smoother on the finest level when the Krylov subspace dimension is limited and such smoother would be able to march in pseudo time with large strides since the stability can be improved with a pMG configuration.…”

Efficient $p$-multigrid method based on an exponential time discretization for compressible steady flows

“…( 6). While the second-order PCEXP scheme is more appropriate for computing unsteady problems, the EXP1 scheme which is the first-order PCEXP scheme is shown to be especially effective for steady flows as in reference [1,3].…”

“…To provide a better smoother with stronger damping effects, the EXP1 scheme that exhibits fast convergence rates for Euler and Navier-Stokes equations is considered. Unlike the explicit RK smoother that only produces weak damping effects in a local, point-wise manner, the exponential scheme is a global method that allows large time steps with strong damping effects to all the frequency modes across the computational domain, as shown in the previous works [3].…”

“…Unfortunately, in contrast to a relative ubiquity of spatial discretizations, efficient time-marching approaches seem to be limited. Recently, as an alternative to traditional time-marching methods, an exponential time integration scheme, predictorcorrector exponential time-integrator scheme (PCEXP) [1][2][3], is developed and successfully applied to the time stepping of CFD problems, exhibiting some advantages in terms of accuracy and efficiency for solving the fluid dynamics problems governing by the Euler and Navier-Stokes equations for either time-dependent and time-independent regimes.…”

Krylov SSP Integrating Factor Runge–Kutta WENO Methods

“…Exponential integrators such as integrating factor methods are time schemes which have large linear stability regions. They can achieve very large time-step sizes or Courant-Friedrichs-Lewy (CFL) numbers for problems with smooth solutions [13][14][15]37]. However, these schemes do not satisfy the SSP property.…”