A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows

Blom, David; Birken, Philipp; Bijl, H.; Kessels, Fleur; Meister, Andreas; Zuijlen, A.H. van

doi:10.1007/s10444-016-9468-x

Cited by 26 publications

(24 citation statements)

References 46 publications

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“…Such behavior can be observed across a range of test problems [16,17,15] as well as for problems that stem from more realistic physical models [7,2,18]. As the before mentioned work shows, this behavior is not limited to one class of numerical method but can be observed for implicit Runge-Kutta methods, BDF methods, implicit-explicit (IMEX) methods, and exponential integrators.…”

Section: Introductionmentioning

confidence: 65%

An adaptive step size controller for iterative implicit methods

Einkemmer

2018

Applied Numerical Mathematics

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The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used.In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion-advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen-Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.

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Section: Introductionmentioning

confidence: 65%

An adaptive step size controller for iterative implicit methods

Einkemmer

2018

Applied Numerical Mathematics

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“…Note that the solution error at each time step size has been determined by performing a preliminary simulation with a very low nonlinear tolerance of 10 −12 and by computing the L 2 -norm of the pointwise differences between the numerical density field and the exact one. For the viscous shock test problem, the linear-solver tolerance was fixed to 10 −2 , and the outer Newton iterations were stopped when the relative decrease from the initial Newton guess of the norm of the right-hand side of Equation (11) was less than 10 −2 . Furthermore, for the first three test cases, the unknown initial solutions needed to start the MEBDF multistep scheme are obtained evaluating the exact solution at the appropriate time levels.…”

Section: Resultsmentioning

confidence: 99%

“…Furthermore, for the first three test cases, the unknown initial solutions needed to start the MEBDF multistep scheme are obtained evaluating the exact solution at the appropriate time levels. For the laminar vortex shedding behind a cylinder, the linear-solver tolerance was fixed to 10 −1 and the Newton algorithms were stopped when the relative decrease from the initial Newton guess of the norm of the right-hand side of Equation (11) was less than 10 −2 . Furthermore, the maximum number of nonlinear iterations was set to 4 to provide an alternative termination criterion to Newton loop.…”

Section: Resultsmentioning

confidence: 99%

“…Recently, a novel approach to high order implicit time discretizations of high‐order DG schemes has been proposed for incompressible and compressible Navier‐Stokes equations. Furthermore, several high‐order semi‐implicit multistage (Rosenbrock), implicit mutlistage multistep (MEBDF, TIAS), and implicit multistep second derivative time integration schemes have been recently proposed in the DG context. Although these schemes allow to overcome the strong restriction on the time step size typical of explicit methods, their use require to compute, store, and factorize large Jacobian matrices.…”

Section: Introductionmentioning

confidence: 99%

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Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Nigro

Bartolo

Crivellini

et al. 2018

Numerical Methods in Fluids

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Summary In this work, a time‐accurate integration of the discontinuous Galerkin space‐discretized Navier‐Stokes equations is performed exploiting the matrix‐free (MF) approach to speed up the solution process of the modified extended backward differentiation formulae (MEBDF) schemes. MEBDF are high‐order accurate implicit multistep schemes composed by three nonlinear stages. The proposed algorithm consists in solving the resulting nonlinear system of each stage with a preconditioned MF Newton/Krylov method using a frozen preconditioner strategy to improve its efficiency. Numerical results for compressible inviscid and viscous test cases, both with a known analytical solution, aim at assessing the performance of the proposed MF‐MEBDF algorithm, by comparing it with the one obtained by using its matrix‐explicit counterpart or the explicit strong stability preserving Runge‐Kutta scheme. In particular, the influence of some relevant physical (low‐speed flows) and discretization (aspect ratio, polynomial degree) aspects on the performance of the different time integration schemes are investigated, highlighting the pros and cons of the proposed algorithm and its effectiveness in solving nonstiff and stiff systems. Furthermore, the scalability in parallel computations of the proposed algorithm is investigated and, finally, its potential for efficient long‐time simulations is demonstrated computing a laminar vortex shedding behind a circular cylinder at different Reynolds numbers and by comparing its effectiveness with that of the strong stability preserving Runge‐Kutta scheme.

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“…In fact, the performance of ROW and ESDIRK highly depends on how the linear and nonlinear systems are solved. Blom et al [45] have shown that ROW is not necessarily more efficient than ESDIRK when a second-order central finite volume method is used. Liu et al [42] has conducted a comparative study of several thirdorder ROW methods and a third-order ESDIRK (ESDIRK3) [21] method with a third-order hierarchical WENO (weighted essentially non-oscillatory) reconstructed discontinuous Galerkin (rDG) method.…”

Section: Introductionmentioning

confidence: 99%

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

Wang

2020

J Sci Comput

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We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) solution of the unsteady Navier-Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of 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 the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

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A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows

Cited by 26 publications

References 46 publications

An adaptive step size controller for iterative implicit methods

An adaptive step size controller for iterative implicit methods

Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

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