A fully conservative mimetic discretization of the Navier–Stokes equations in cylindrical coordinates with associated singularity treatment

Oud, Guido; Heul, D. R. van der; Vuik, C.; Henkes, R.A.W.M.

doi:10.1016/j.jcp.2016.08.038

Cited by 4 publications

(5 citation statements)

References 14 publications

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“…A generalization of his approach to non-uniform grids was presented by Vasilyev [139] and Ham et al [49]. As a curvilinear case, the energy-preserving formulation in cylindrical coordinates was studied in [40,79,87]; a more general approach for structured curvilinear staggered grids has been proposed in [138]. Discrete skewsymmetry of the convective terms also features in the summation-by-parts (SBP) method introduced by Strand [123] and Olsson [84,85], and generalized in [74,81,127,128].…”

Section: Introductionmentioning

confidence: 99%

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

Veldman

2021

SIAM Rev.

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Section: Introductionmentioning

confidence: 99%

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

Veldman

2021

SIAM Rev.

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“…The flux vectors F e = (F e,j ) j and F n = (F n,k ) k are used to calculate the discrete operator DIV corresponding to equation (23). Standard differentiation stencils are applied to calculate DIV (which lives on the pressure points).…”

Section: Linear-wave Equations: Operators DIV and Gradmentioning

confidence: 99%

“…A higher-order finite-difference approximation DIV for the divergence is constructed from an exact formulation (23) of the divergence in a cell center. Subsequently, standard finite-difference stencils are applied to this formulation, resulting in a higher-order, conservative approximation of the divergence.…”

Section: Linear-wave Equations: Operators DIV and Gradmentioning

confidence: 99%

“…Mimetic finite-difference methods also mimic the important properties of differential operators. An interesting review is given in [21], and recently, a second-order mimetic discretization of the Navier-Stokes equations conserving mass, momentum, and kinetic energy was presented in [23].…”

Section: Introductionmentioning

confidence: 99%

“…The combination of a symmetry-preserving discretization and regularization for compressible flows is studied in [27].Mimetic finite-difference methods also mimic the important properties of differential operators. An interesting review is given in [21], and recently, a second-order mimetic discretization of the Navier-Stokes equations conserving mass, momentum, and kinetic energy was presented in [23].Castillo et al have provided a framework for mimetic operators in [7,8]. A second-and fourthorder mimetic approach is constructed for non-uniform rectilinear staggered meshes in [1,3,9].…”

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confidence: 99%

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Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids

Hof

Vuik

2019

Journal of Computational Science

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Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and energy are proven in the same way as for the original continuous model. This paper presents a new finite-difference symmetry-preserving space discretization. Boundary conditions and time integration are not addressed. The novelty is that it combines arbitrary order of convergence, orthogonal and non-orthogonal structured curvilinear staggered meshes, and the applicability to a wide variety of continuous operators, involving chain rules and nonlinear advection, as illustrated by the shallow-water equations. Experiments show exact conservation and convergence corresponding to expected order.In [42,43], a fourth-order symmetry-preserving finite-volume method is constructed using Richardson extrapolation of a second-order symmetry-preserving method [40]. The extension to unstructured collocated meshes is presented in [32], and an application can be seen in [41]. The extension to upwind discretizations was made in [39], and a discretization for the advection operator for curvilinear collocated meshes was found in [16]. In [17], the method is extended to non-uniform structured curvilinear collocated grids by deriving a discrete product rule. Furthermore, a symmetry-preserving method that conserves mass and energy for compressible-flow equations with a state equation is described in [35]. For rectilinear grids, this method works well, but it is challenging to let this method work for unstructured grids [35]. Finally, in [26], a symmetry-preserving discretization for curvilinear collocated and rectilinear staggered meshes is found exploiting the skew-symmetric nature of the advection operator on square-root variables.Another option to preserve symmetry is to use discrete filters to regularize the convective terms of the equation [33,19]. The combination of a symmetry-preserving discretization and regularization for compressible flows is studied in [27].Mimetic finite-difference methods also mimic the important properties of differential operators. An interesting review is given in [21], and recently, a second-order mimetic discretization of the Navier-Stokes equations conserving mass, momentum, and kinetic energy was presented in [23].Castillo et al. have provided a framework for mimetic operators in [7,8]. A second-and fourthorder mimetic approach is constructed for non-uniform rectilinear staggered meshes in [1,3,9]. In [10] their method is extended to curvilinear staggered meshes, but discrete conservation of mass, momentum and energy is not shown. A second-order mimetic finite-difference method for rectilinear staggered meshes is also constructed in [28].Other mimetic finite-difference methods use algebraic topology to design and analyze compatible discrete operators corresponding to a continuous formulation [4,18,25]. In order to construct a discrete de Rham complex, certain conditions...

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Numerical treatment of incompressible turbulent flow

Verstappen

Veldman

2023

Numerical Methods in Turbulence Simulation

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A fully conservative mimetic discretization of the Navier–Stokes equations in cylindrical coordinates with associated singularity treatment

Cited by 4 publications

References 14 publications

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids

Numerical treatment of incompressible turbulent flow

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