On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation

Gary K. Yan; Carl Ollivier-Gooch

doi:10.1115/1.4039188

Journal of Verification, Validation and Uncertainty Quantification | Volume 2 | Issue 4 | research-article

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On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation

Gary K. Yan and Carl Ollivier-Gooch

[+] Author and Article Information

Gary K. Yan

Department of Mechanical Engineering,
The University of British Columbia,
Vancouver, BC V6T 1Z4, Canada
e-mail: gary.yan@alumni.ubc.ca

Carl Ollivier-Gooch

Professor
Fellow ASME
Department of Mechanical Engineering,
The University of British Columbia,
Vancouver, BC V6T 1Z4, Canada
e-mail: cfog@mech.ubc.ca

Manuscript received May 3, 2016; final manuscript received January 10, 2018; published online February 23, 2018. Editor: Ashley F. Emery.

J. Verif. Valid. Uncert 2(4), 041003 (Feb 23, 2018) (17 pages) Paper No: VVUQ-16-1013; doi: 10.1115/1.4039188 History: Received May 03, 2016; Revised January 10, 2018

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Abstract

A numerical estimation of discretization error for steady compressible flow solutions is performed using the error transport equation (ETE). There is a deficiency in the literature for obtaining efficient, higher order accurate error estimates for finite volume discretizations using nonsmooth unstructured meshes. We demonstrate that to guarantee sharp, higher order accurate error estimates, one must discretize the ETE to a higher order than the primal problem, a requirement not necessary for uniform meshes. Linearizing the ETE can limit the added cost, rendering the overall computational time competitive, while retaining accuracy in the error estimate. For the Navier–Stokes equations, when the primal solution is corrected using this error estimate, for the same level of solution accuracy the overall computational time is more than two times faster compared to solving the higher order primal problem. In addition, our scheme has robustness advantages, because we solve the primal problem only to lower order.

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Figures

Fig. 1

Comparison of truncation error and estimate by higher order reconstruction of the primal solution of the Poisson problem using periodic boundary conditions: (a) Exact truncation error for the second-order scheme and its estimate using a fourth order reconstruction of the second-order solution, on a uniform mesh, (b) exact truncation error minus the estimate for the uniform mesh. Note the smoothness and small magnitude. (c) Exact truncation error for the second-order scheme and its estimate using a fourth order reconstruction of the second-order solution, on a perturbed mesh, and (d) exact truncation error minus the estimate for the perturbed mesh. Note the large amplitude noise.

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Fig. 2

Convergence of the error estimate with mesh length scale for linear and nonlinear 1D problems: (a) advection equation, (b) diffusion equation, and (c) viscous Burgers' equation

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Fig. 3

Convergence in error estimate with mesh length scale using the fully nonlinear error flux, the linearized flux, and the primal flux for the 1D viscous Burgers' equation: (a) the (2, 4, 4) scheme and (b) the (2, 6, 6) scheme

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Fig. 4

Discretization error and estimate for the nonlinear and linearized ETE using the (2, 4, 4) scheme for the 1D viscous Burgers' equation

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Fig. 5

Exact error, error estimate, and difference for the steady advection problem in 2D using the (2, 4, 4) discretization: (a) exact error, (b) (2, 4, 4) error estimate, and (c) difference between exact error and estimate

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Fig. 6

Convergence of the error estimate with mesh length scale for linear and nonlinear problems in 2D: (a) advection equation, (b) diffusion equation, and (c) viscous Burgers' equation

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Fig. 7

Exact error, error estimate, and difference for the steady diffusion problem in 2D using the (2, 4, 4) discretization: (a) exact error, (b) (2, 4, 4) error estimate, and (c) difference between exact error and estimate

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Fig. 8

Exact error and estimates using the (2, 4, 4) scheme for viscous Burgers' equation in 2D: (a) exact error, (b) (2, 4, 4) error estimate, and (c) (2, 4, 4) linearized error estimate

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Fig. 9

Computed solution, and differences in error using the (2, 4, 4) schemes for viscous Burgers' equation: (a) computed solution, (b) difference in error estimate using (2, 4, 4), and (c) difference in error estimate using linearized (2, 4, 4)

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Fig. 10

Comparison of linearized ETE for 2D viscous Burgers' Equation. The nonlinear and linearized ETE are almost indistinguishable.

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Fig. 11

Exact error and estimates using the (2, 4, 4) scheme for the supersonic vortex test case: (a) exact error, (b) (2, 4, 4) error estimate, and (c) linearized (2, 4, 4) error estimate

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Fig. 12

Computed solution, and differences in error using the (2, 4, 4) schemes for the supersonic vortex test case: (a) computed solution (energy), (b) difference in error estimate using (2, 4, 4), and (c) difference in error estimate using linearized (2, 4, 4)

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Fig. 13

Convergence of the error estimate with mesh length scale for the supersonic vortex problem

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Fig. 14

Exact error and estimates using the (2, 4, 4) scheme for 2D Navier–Stokes equations: (a) exact error, (b) (2, 4, 4) error estimate, and (c) linearized (2, 4, 4) error estimate

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Fig. 15

Computed solution, and differences in error using the (2, 4, 4) schemes for 2D Navier–Stokes equations: (a) computed solution (energy), (b) difference in error estimate using (2, 4, 4), and (c) difference in error estimate using linearized (2, 4, 4)

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Fig. 16

Convergence of the error estimate with mesh length scale for the 2D Navier–Stokes equations

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Fig. 17

Computed solution and error estimates for flow around the NACA 0012 airfoil: (a) computed solution (energy), (b) (2, 4, 4) error estimate, and (c) linearized (2, 4, 4) error estimate

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Fig. 18

Comparison of solution error plotted against overall computational time required for the 2D Navier–Stokes equations using the manufactured solution test case, using freestream and continuous exact initial conditions. The (4, 0, 0) scheme did not converge for the finest mesh using freestream initial conditions. (a) Time comparison using freestream initial conditions and (b) time comparison using continuous exact initial conditions.

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Fig. 19

Comparison of an output functional (C_D_{, Viscous}) plotted against mesh length scale and overall computational time required for 2D Navier–Stokes equations for the NACA 0012 test case: (a) output functional (C_D_{, Viscous}) against mesh length scale and (b) output functional (C_D_{, Viscous}) against mesh overall computational time

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Fig. 20

Particularly unsuitable meshes which converge for the linearized (2, 4, 4) scheme but not (4, 0, 0) or (2, 4, 4): (a) elliptic mesh in which the wake region is not properly resolved and (b) hyperbolic mesh in which the boundary layer region is not properly resolved

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Yan GK, Ollivier-Gooch C. On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation. ASME. J. Verif. Valid. Uncert. 2018;2(4):041003-041003-17. doi:10.1115/1.4039188.

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On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation

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