A comparison of the efficiency of Rosenbrock and DIRK variants (original) (raw)
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.
High-Order Implicit Time Integration for Unsteady Compressible Fluid Flow Simulation
2013
This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analysis and optimization capabilities of the framework are verified by rederiving methods with known coefficients. The framework is then applied to the derivation of novel singly-diagonally-implicit Runge-Kutta methods, explicit-first-stage singly-diagonally implicit Runge-Kutta methods, and singly-diagonallyimplicit multistep Runge-Kutta methods. The fourth-order methods developed have similar efficiency to contemporary methods; however a fifth-order explicit-first-stage singlydiagonally-implicit Runge-Kutta method is obtained with higher relative efficiency. This is confirmed with simulations of ...
High-order implicit time-marching methods for unsteady fluid flow simulation
2015
HIGH-ORDER IMPLICIT NUMERICAL METHODS FOR UNSTEADY FLUID FLOW SIMULATION Pieter D. Boom <pieter.boom@mail.utoronto.ca> Doctor of Philosophy Graduate Department of Aerospace Science and Engineering University of Toronto 2015 Unsteady computational fluid dynamics (CFD) is increasingly becoming a critical tool in the development of emerging technologies and modern aircraft. In spite of rapid mathematical and technological advancement, these simulations remain computationally intensive and time consuming. More efficient temporal integration will promote a wider use of unsteady analysis and extend its range of applicability. This thesis presents an investigation of efficient high-order implicit time-marching methods for application in unsteady compressible CFD. A generalisation of time-marching methods based on summation-by-parts (SBP) operators is described which reduces the number of stages required to obtain a prescribed order of accuracy, thus improving their efficiency. The cl...
High Order Finite Volume Schemes for Numerical Solution of Unsteady Flows
The aim of this contribution is to present two modern high-order finite volume (FVM) schemes for numerical solution of unsteady transonic flows. The first one is derived from the total variation diminishing (TVD) version of the classical MacCormack scheme proposed by Causon. In our case it is used with slight modifications and hence refered to as Modified Causon’s scheme. It is no more TVD, but with no loss of accuracy to the TVD version and with a significantly lower demands on computational power and memory (cca 30% less). The second one, based on a similar approach as the WENO family schemes, is the implicit Weighted Least-Square Reconstruction scheme (WLSQR) used in combination with the AUSMPW+ numerical flux. For the turbulence modelling the Kok’s TNT turbulence model is employed. Unsteady effects (forced oscillatory motion) are simulated by Arbitrary Lagrangian–Eulerian method (ALE). As the transonic test cases the inviscid and turbulent flow around the NACA 0012 profile and inviscid flow over the ONERA M6 wing were chosen. Comparison of numerical and experimental results for inviscid flow is very good, which is unfortunately not the case of turbulent flow.
International Journal for Numerical Methods in Fluids
This article develops high-order implicit time-stepping methods combined with the fourth-order central essentially-non-oscillatory (CENO) scheme for stiff three-dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three-dimensional multiblock body-fitted grids with hexahedral cells. Several implicit time integration methods of third-and fourth-order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge-Kutta (ESDIRK4), and Rosenbrock-type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian-free inexact Newton-Krylov method with additive Schwarz preconditioning using block-based incomplete LU decomposition. The performance of the high-order implicit time-stepping methods on smooth and stiff problems is compared with a standard fourth-order explicit Runge-Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth-order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth-order CENO method. When applied to the problem of vortex shedding governed by the Navier-Stokes equations, an A-stable ESDIRK4 scheme proved to be the more robust and accurate implicit time-marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high-order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high-order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high-order
International Journal of Computational Science and Engineering, 2014
In the numerical simulation of inviscid and viscous compressible fluid flow, implicit Newton-Krylov methods are frequently used. A crucial ingredient of Krylov subspace methods is the evaluation of the product of the Jacobian matrix of the spatial operator, e.g., fluxes, and a Krylov vector. In this article we consider a matrix-free implementation of the Jacobian-vector product within the flow solver QUADFLOW using automatic differentiation. The convergence of the nonlinear iteration using first-and second-order accurate Jacobian-vector products is compared. It turns out that a hybrid implementation employing both, first-and secondorder accurate methods, significantly reduces the overall execution time of the simulation.