On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

Roy, Christopher J.; Sinclair, Andrew J.

doi:10.1016/j.jcp.2008.11.008

Cited by 23 publications

(14 citation statements)

References 11 publications

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“…Nonetheless, at least for the unforced problem of Section 5.2, imposing physical boundary conditions accurately and stably at high Reynolds numbers is difficult, especially for the case of traction boundary conditions. To demonstrate the capability of the scheme for problems with characteristics of high Re flows (e.g., thin boundary layers), in Section 5.3 we also present steady solutions obtained by the method for the wellknown lid-driven cavity flow, along with a regularized version [36] of this problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

Griffith

2009

Journal of Computational Physics

119

211

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Section: Numerical Resultsmentioning

confidence: 99%

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

Griffith

2009

Journal of Computational Physics

119

211

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“…The weighting function-based MNP approach described earlier used sophisticated spline fits which allowed for arbitrary levels of continuity across spline zone boundaries and could be readily extended to multiple dimensions. 21 However, the generation of such global functions is expensive and can lead to oscillations in regions with insufficient resolution. When using MNP/defect correction for discretization error estimation purposes, global fits are likely not necessary, so the use of local fits will also be explored.…”

Section: Continuous Defect Correctionmentioning

confidence: 99%

“…(21) The flow field is defined as a function of variables at the inner radius of the annulus denoted by the subscript i. For the problems examined herein, the inner radius, , is 2.0, the outer radius is 3.0, the inner density, , is 1.0 kg/m 3 , and the inner Mach number, , is 2.0.…”

Section: Supersonic Vortexmentioning

confidence: 99%

Discretization error estimation and exact solution generation using the method of nearby problems.

Sinclair

Raju

Kurzen

et al. 2011

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The Method of Nearby Problems (MNP), a form of defect correction, is examined as a method for generating exact solutions to partial differential equations and as a discretization error estimator. For generating exact solutions, four-dimensional spline fitting procedures were developed and implemented into a MATLAB code for generating spline fits on structured domains with arbitrary levels of continuity between spline zones. For discretization error estimation, MNP/defect correction only requires a single additional numerical solution on the same grid (as compared to Richardson extrapolation which requires additional numerical solutions on systematically-refined grids). When used for error estimation, it was found that continuity between spline zones was not required. A number of cases were examined including 1D and 2D Burgers' equation, the 2D compressible Euler equations, and the 2D incompressible Navier-Stokes equations. The discretization error estimation results compared favorably to Richardson extrapolation and had the advantage of only requiring a single grid to be generated.

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“…The definition of the discretization error is u u h h− = ε (17) and thus we can rewrite Equation (16) as…”

Section: Relationship Between Truncation Error and Discretization mentioning

confidence: 99%

“…16 This spline fitting approach has been used to develop continuous approximations of 2D heat transfer and Navier-Stokes CFD simulations by Roy and Sinclair 17 and is referred to as the Method of Nearby Problems (MNP). The MNP curve fitting procedure involves breaking the numerical solution up into overlapping zones that are approximated with local least squares polynomial fits.…”

mentioning

confidence: 99%

Strategies for Driving Mesh Adaptation in CFD (Invited)

Roy

2009

47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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This paper examines different approaches for driving mesh adaptation and provides theoretical developments for understanding the relationship between discretization error, the numerical scheme, and the mesh. Discrete and continuous equations governing the transport of discretization error are developed and it is shown that the truncation error acts as the local source for these equations. Examination of the truncation error in generalized coordinates provides insight into the role of mesh quality (mesh stretching for the 1D case) in the discretization error. Numerical results are presented for 1D steady-state Burgers equation at Reynolds numbers of 32 and 128. Four different approaches for driving mesh adaption are implemented for this case: solution gradients, solution curvature, discretization error, and truncation error. The truncation-error based adaption is shown to provide superior results for both cases. Finally, two approaches for estimating the truncation error are also discussed which would allow truncation error-based adaption to be implemented for complex numerical methods.

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On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

Cited by 23 publications

References 11 publications

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

Discretization error estimation and exact solution generation using the method of nearby problems.

Strategies for Driving Mesh Adaptation in CFD (Invited)

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