Analysis and development of compact finite difference schemes with optimized numerical dispersion relations

Kuo, Y.J.; Lee, Long; Lyng, Gregory

doi:10.48550/arxiv.1409.3535

Cited by 1 publication

(1 citation statement)

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“…Examples of interpolation base functions are shown in Figure 1. The grid wavenumber ω corresponds to the number of grid points per wavelength in the grid [8,12]. For a given problem, which has a given Fourier spectrum, refining the grid has the effect of reducing the grid wavenumber.…”

Section: Interpolation Functions On a Uniform 1d Gridmentioning

confidence: 99%

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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Section: Interpolation Functions On a Uniform 1d Gridmentioning

confidence: 99%