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Research Papers

Numerical Errors in Unsteady Flow Simulations

[+] Author and Article Information
L. Eça

Mechanical Engineering Department,
Instituto Superior Técnico,
ULisboa,
Avenida Rovisco Pais 1,
Lisboa 1049 001, Portugal
e-mail: luis.eca@ist.utl.pt

G. Vaz

Maritime Research Institute Netherlands,
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl

S. L. Toxopeus

Maritime Research Institute Netherlands,
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: S.Toxopeus@marin.nl

M. Hoekstra

Voorthuizen 3781XK, The Netherlands

1Corresponding author.

Manuscript received January 9, 2019; final manuscript received June 7, 2019; published online July 11, 2019. Assoc. Editor: William Rider.

J. Verif. Valid. Uncert 4(2), 021001 (Jul 11, 2019) (10 pages) Paper No: VVUQ-19-1001; doi: 10.1115/1.4043975 History: Received January 09, 2019; Revised June 07, 2019

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

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Copyright © 2019 by ASME
Topics: Errors , Simulation
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References

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Eça, L. , and Hoekstra, M. , 2009, “ Evaluation of Numerical Error Estimation Based on Grid Refinement Studies With the Method of the Manufactured Solutions,” Comput. Fluids, 38(8), pp. 1580–1591. [CrossRef]
Eça, L. , Vaz, G. , and Hoekstra, M. , 2018, “ On the Role of Iterative Errors in Unsteady Flow Simulations,” Numerical Towing Tank Symposium (NuTTS), Cortona, Italy, Sept. 30–Oct. 2. https://www.researchgate.net/publication/328064499_On_the_Role_of_Iterative_Errors_in_Unsteady_Flow_Simulations
Eça, L. , and Hoekstra, M. , 2014, “ A Procedure for the Estimation of the Numerical Uncertainty of CFD Calculations Based on Grid Refinement Studies,” J. Comput. Phys., 262, pp. 104–130. [CrossRef]
Eça, L. , Klaij, C. M. , Vaz, G. , Hoekstra, M. , and Pereira, F. S. , 2016, “ On Code Verification of RANS Solvers,” J. Comput. Phys., 310(C), pp. 418–439. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Number of cycles Ncycles required to obtain |ϕ(t)−ϕ(t+T)|<10−10 as a function of the iterative convergence criterion εit and grid/time refinement ratio λi. Comax corresponds to the maximum Courant number. MSP periodic manufactured solution.

Grahic Jump Location
Fig. 2

L norm of the numerical error at half the cycle (enum(u)t=999.5) as a function of the grid/time refinement ratio λi for different values of the iterative convergence criterion εit. Comax corresponds to the maximum Courant number. MSP periodic manufactured solution.

Grahic Jump Location
Fig. 3

L norm of the numerical error at t = 5 (enum(u)t=5) as a function of the grid/time refinement ratio λi for different values of the iterative convergence criterion εit. Comax corresponds to the maximum Courant number. MST transient manufactured solution.

Grahic Jump Location
Fig. 4

Iterative error (eit) at x=0.5,t=999.5 for MSP and x=0.5,t=5 for MST as a function of the iterative convergence criterion εit for different values of the grid/time refinement ratio λi. Comax corresponds to the maximum Courant number. MSP and MST manufactured solutions.

Grahic Jump Location
Fig. 5

Average number of iterations Niter performed for the time steps of the last simulated cycle as a function of the grid/time refinement ratio λi for different values of the iterative convergence criterion εit. Comax corresponds to the maximum Courant number. MSP periodic manufactured solution.

Grahic Jump Location
Fig. 6

Ratio between the estimated iterative error based on an exponential decay and the iterative error obtained from a solution converged to machine accuracy, Rit=eoit/eit. Variables u1, u2, u3, and Ix at all time steps of the last simulated cycle of MSP and for 0<t≤20 of MST. Ffail⇒Rit<1 and Fok⇒1≤Rit≤5. εit corresponds to the lowest iterative convergence criterion used to estimate eoit. Comax is the maximum Courant number. MSP and MST manufactured solutions.

Grahic Jump Location
Fig. 7

Ratio between the estimated discretization error and the “exact” discretization error Rd=(ϕi−ϕo)/(ϕi−ϕexact) for u1, u2, u3, and Ix at t =999.5 for MSP and t = 5 for MST. D1–D4 correspond to the groups of grids/time steps given in Table 2 and SR stands for simultaneous grid/time refinement, IR is independent grid/time refinement, and AR is arbitrary grid/time refinement.

Grahic Jump Location
Fig. 8

Estimation of discretization error based on Eqs. (4) and (5) for u1 at t =999.5 for MSP and u3 at t = 5 for MST. D1–D4 correspond to the groups of grids/time steps given in Table 2. SR stands for simultaneous grid/time refinement and IR corresponds to grid refinement for the smallest time-step plus time refinement for the finest grid.

Grahic Jump Location
Fig. 9

Number of cases that exhibit enum>eit2+ed2 as a function of the grid/time refinement ratio λi for different values of the iterative convergence criterion εit. Comax corresponds to the maximum Courant number. MSP and MST manufactured solutions.

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