Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes

Hoffman, Joe D.

doi:10.1016/0021-9991(82)90028-6

Cited by 63 publications

(33 citation statements)

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“…We remark that we have exploited the above representation in order to make explicit the dependence on the numerical derivatives inside the discretization (15)- (16). We proceed similarly for S 2 and, recalling (23), we get…”

Section: Error Estimates For First Order Schemesmentioning

confidence: 99%

“…with L F + and L F − the Lipschitz constants of numerical functions (15)- (16). By applying to (34) the Hölder's inequality, for 1 ≤ p < +∞, we obtain that…”

Section: Error Estimates For First Order Schemesmentioning

confidence: 99%

“…that is derived from (9) for appropriate weighted interfacial jumps (refer also to [16], [8], [30], [21] and [18]). A straightforward computation shows the truncation error vanishing for nonuniform grids, thus inducing the correct convergence.…”

Section: Empirical Analysismentioning

confidence: 99%

“…Within the context of balance laws, a first attempt at demonstrating error estimates for (finite volume) approximations on uniform spatial meshes of the source term (3) is made in [19]. But elementary counter-examples show that (strong) convergence commonly fails for nonuniform grids, since standard consistency conditions do not guarantee the truncation errors vanishing (refer to [16], [27], [30] and [18] for more details). Therefore, some explicit dependency of the discretization upon the cells' size is necessary to an error analysis with optimal rates (see [8], [2], and the references therein).…”

Section: Introductionmentioning

confidence: 99%

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Three-points interfacial quadrature for geometrical source terms on nonuniform grids

Katsaounis

Simeoni

2011

Calcolo

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Section: Error Estimates For First Order Schemesmentioning

confidence: 99%

“…with L F + and L F − the Lipschitz constants of numerical functions (15)- (16). By applying to (34) the Hölder's inequality, for 1 ≤ p < +∞, we obtain that…”

Section: Error Estimates For First Order Schemesmentioning

confidence: 99%

Section: Empirical Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

Katsaounis

Simeoni

2011

Calcolo

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“…For structured grids several truncation error estimators have been proposed for particular discretisation schemes, for example in [1], [2] and [3]. They express the leading term of the truncation error in terms of the derivatives of the flow variables and the geometry of the grid.…”

Section: Introductionmentioning

confidence: 99%

Estimate of the truncation error of finite volume discretization of the Navier-Stokes equations on colocated grids

Syrakos

Goulas

2005

Int. J. Numer. Meth. Fluids

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A methodology is proposed for the calculation of the truncation error of finite volume discretisations of the incompressible Navier -Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier -Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretisations of the incompressible Navier -Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure-part of the mass fluxes is not dependent on the coefficients of the linearised momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two-dimensional flows, but extension to three-dimensional cases should not pose problems.

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One-dimensional stretching functions for Cn patched grids, and associated truncation errors in finite-difference calculations

Nitsche

1996

Commun. Numer. Meth. Engng.

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SUMMARYIn this work the truncation-error criteria of are combined with conditions of vanishing second and higher derivatives at both endpoints for the purpose of deriving new classes of onedimensional stretching functions for mesh refinement in finite-difference numerics. With these elementary stretching functions, matching of the slopes between adjacent grid patches then automatically confers C" regularity upon the composite stretching function. Formulated with reference to two conceptions of truncation order (fixed relative distribution against fixed number of nodes) the resulting mappings are shown to provide particularly advantageous node distributions at both ends simultaneously (with concomitantly higher truncation error in between). Viewed overall, the truncation-error functions compare favourably with those for sinh, tanh and erf -mappings whose utility for mesh refinement was established by Thompson and Mastin. The numerical labour of implementing the new stretching functions is only slightly greater than that required for the error function. An illustrative derivation involving c" patching leads to two-sided stretching functions, which allow the slopes at both ends to be prescribed arbitrarily. This formulation differs from a previous approach described by Vinokur (1983).

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Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes

Cited by 63 publications

References 1 publication

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

Estimate of the truncation error of finite volume discretization of the Navier-Stokes equations on colocated grids

One-dimensional stretching functions for Cn patched grids, and associated truncation errors in finite-difference calculations

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