A comparison of the efficiency of Rosenbrock and DIRK variants (original) (raw)
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Rosenbrock time integration for unsteady flow simulations
This contribution compares the efficiency of Rosenbrock time integration schemes with ESDIRK schemes, applicable to unsteady flow and fluid-structure interaction simulations. Compared to non-linear ESDIRK schemes, the linear implicit Rosenbrock-Wanner schemes require subsequent solution of the same linear systems with different right hand sides. By solving the linear systems with the iterative solver GMRES, the preconditioner can be reused for the subsequent stages of the Rosenbrock-Wanner scheme. Unsteady flow simulations show a gain in computational efficiency of approximately factor three to five in comparison with ESDIRK.
Time integration schemes for the unsteady Navier-Stokes equations
15th AIAA Computational Fluid Dynamics Conference, 2001
The e ciency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the e ciency of higher-order Runge-Kutta schemes in comparison with the popular Backward Di erencing Formulations. For this comparison an unsteady two-dimensional laminar ow problem is chosen, i.e. ow around a circular cylinder at Re=1200. It is concluded that for realistic error tolerances smaller than 10 ,1 fourth-and fth-order Runge Kutta schemes are the most e cient. For reasons of robustness and computer storage, the fourth-order Runge-Kutta method is recommended. The e ciency of the fourth-order Runge-Kutta scheme exceeds that of second-order Backward Di erence Formula BDF2 by a factor of 2:5 at engineering error tolerance levels 10 ,1 -10 ,2 . E ciency gains are more dramatic at smaller tolerances.
A comparison of time integration methods in an unsteady low-Reynolds-number flow
International Journal for Numerical Methods in Fluids, 2002
This paper describes three di erent time integration methods for unsteady incompressible Navier-Stokes equations. Explicit Euler and fractional-step Adams-Bashford methods are compared with an implicit three-level method based on a steady-state SIMPLE method. The implicit solver employs a dual time stepping and an iteration within the time step. The spatial discretization is based on a co-located ÿnitevolume technique. The in uence of the convergence limits and the time-step size on the accuracy of the predictions are studied. The e ciency of the di erent solvers is compared in a vortex-shedding ow over a cylinder in the Reynolds number range of 100-1600. A high-Reynolds-number ow over a biconvex airfoil proÿle is also computed. The computations are performed in two dimensions. At the low-Reynolds-number range the explicit methods appear to be faster by a factor from 5 to 10. In the high-Reynolds-number case, the explicit Adams-Bashford method and the implicit method appear to be approximately equally fast while yielding similar results. Copyright ? 2002 John Wiley & Sons, Ltd.
A global time integration approach for realistic unsteady flow computations
54th AIAA Aerospace Sciences Meeting, 2016
A novel time integration approach is explored for unsteady flow computations. It is a multi-block formulation in time where one solves for all time levels within a block simultaneously. The time discretization within a block is based on the summation-by-parts (SBP) technique in time combined with the simultaneous-approximation-term (SAT) technique for imposing the initial condition. The approach is implicit, unconditionally stable and can be made high order accurate in time. The implicit system is solved by a dual time stepping technique. The technique has been implemented in a flow solver for unstructured grids and applied to an unsteady flow problem with vortex shedding over a cylinder. Four time integration approaches being 2 nd to 5 th order accurate in time are evaluated and compared to the conventional 2 nd order backward difference (BDF2) method and a 4 th order diagonally implicit Runge-Kutta scheme (ESDIRK64). The obtained orders of accuracy are higher than expected and correspond to the accuracy in the interior of the blocks, up to 8 th order accuracy is obtained. The influence on the accuracy from the size of the time blocks is small. Smaller blocks are computationally more efficient though, and the efficiency increases with increased accuracy of the SBP operator and reduced size of time steps. The most accurate scheme, with a small time step and block size, is approximately as efficient as the ESDIRK64 scheme. There is a significant potential for improvements ranging from convergence acceleration techniques in dual time, alternative initialization of time blocks, and by introducing smaller time blocks based on alternative SBP operators.
An exponential time-integrator scheme for steady and unsteady inviscid flows
Journal of Computational Physics
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.
Computers & Fluids, 2019
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Application of Rosenbrock schemes to 3D flow problems with moderate Reynolds numbers • Imposition of non-homogeneous time dependent Dirichlet boundary conditions • Numerical computation of convergence orders • Evaluation of an adaptive timestep strategy in two benchmark test cases
2019
We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit rst stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for unsteady flow simulation using spatially high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) formulations. The pseudo-transient continuation is employed to solve the nonlinear systems resulting from the temporal discretizations with ESDIRK and BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method (GMRES) solver is employed with a low storage element-Jacobi preconditioner to solve linear systems, including those in linearly implicit ROW methods and those from linearization of the nonlinear systems in ESDIRK and BDF2 methods. We observe that all ROW and ESDIRK schemes (from second order to fourth order) are more computationally efficient than BDF2, and ROW methods can potential...
Non-Oscillatory Limited-Time Integration for Conservation Laws and Convection-Diffusion Equations
arXiv (Cornell University), 2022
In this study we consider unconditionally non-oscillatory, high order implicit time marching based on timelimiters. The first aspect of our work is to propose the high resolution Limited-DIRK3 (L-DIRK3) scheme for conservation laws and convection-diffusion equations in the method-of-lines framework. The scheme can be used in conjunction with an arbitrary high order spatial discretization scheme such as 5th order WENO scheme. It can be shown that the strongly S-stable DIRK3 scheme is not SSP and may introduce strong oscillations under large time step. To overcome the oscillatory nature of DIRK3, the key idea of L-DIRK3 scheme is to apply local time-limiters (K.Duraisamy, J.D.Baeder, J-G Liu), with which the order of accuracy in time is locally dropped to first order in the regions where the evolution of solution is not smooth. In this way, the monotonicity condition is locally satisfied, while a high order of accuracy is still maintained in most of the solution domain. For convenience of applications to systems of equations, we propose a new and simple construction of time-limiters which allows flexible choice of reference quantity with minimal computation cost. Another key aspect of our work is to extend the application of time-limiter schemes to multidimensional problems and convection-diffusion equations. Numerical experiments for scalar/systems of equations in one-and two-dimensions confirm the high resolution and the improved stability of L-DIRK3 under large time steps. Moreover, the results indicate the potential of time-limiter schemes to serve as a generic and convenient methodology to improve the stability of arbitrary DIRK methods.